Angel: Angel is fond of playing this role, often as a Hawaiian-Shirted Tourist. On set, the shorthand for this became "Herb Saunders" (Angel's alias in "Sense & Sensitivity").
Lt. Columbo, famously. Maybe the single most well-developed example ever on television, to the point that many doubt it's an act at all.
Teen Wolf: Lydia. She outsmarts Jackson by knowing a cougar and a mountain lion are the same thing, then plays dumb again asking "Isn't it?". She also has a 5.0 GPA. Though by season 3, Jackson is gone, and Lydia is much more willing to display her intelligence.
The Beverly Hillbillies: Jed Clampett, in particular the first seasons, despite his stereotypical "hayseed" appearance and mannerisms, clearly is a lot smarter than he lets on. And any time he is acting particularly dumb, or proclaims his ignorance, he's usually trying to make people think he's Too Dumb to Fool.
Community: Jeff Winger, while obviously very intelligent, rarely chooses to use said intelligence, instead skating by off of other people's efforts.
Green Acres had Eb take an accountancy course via the Clarkwell Institute only to be sent an acting course instead. Turns out Mr. Drucker and Mr. Kimball are also Clarkwell alums (who had also gotten the wrong courses which led them to their current careers). Clarkwell also has its own anthem.
Sledge Hammer!: Used every few episodes by Detective Hammer. He is written off by everybody as a violent idiot, but then manages to crack the case, with a sudden admission that he had been taking notes all along — even if in most other episodes he was just a lucky idiot. This is even given a spot of Lampshade Hanging in one episode, when a game show host killed off a competitor who was getting close to the top prize. He avoided having to do this before by only recruiting idiots. When the sidekick is confused how someone that smart got on the show, Sledge enlightens us with this line: "You see, he looks dumb, but turned out to be a genius; a trick I've used myself."
Cordelia Chase initially appeared to be The Ditz, but a throw-away gag in Season Three revealed she was actually rather bright when she aced her SATs, and a minor sub-plot later in the season further established her academic creds. Her later appearances in Angel followed up on this development, showing her as far more intelligent than she had originally seemed (not that this was difficult).
Buffy herself also personifies this trope. She's got the vapid California blonde act down to perfection and uses it quite frequently as part of her arsenal to get her enemies to underestimate her. However, as one of the longest-lived slayers in history and a combined SAT score of 1430, she's definitely more than a pretty face.
The Doctor, particularly in his second and fourth incarnations, often used to play the fool to lull his enemies into a false sense of security ("Would you care for a jelly baby?"). Later on, in 'Family of Blood' whilst pretending to be still human, his blundering nervousness lulls the Family into such a state of arrogant superiority that they don't notice that the buttons he's 'accidentally' pushing will destroy their ship until it's much, much too late. This was occasionally subverted, as well, with the enemies realising that this was what he was doing - most notably in 'City of Death', in which Count Scarlioni's understanding of the Doctor prevents him from being deceived.
Countess: My dear, I don't think he's as stupid as he seems.
Count Scarlioni: My dear, nobody could be as stupid as he seems.
The Seventh Doctor also has his moments of this, seeing how he often appears to play the buffoon only to continually outwit the enemy without effort. Particularly highlighted with his 'adjustment' of old quotes.
The Doctor: "A stitch in time...takes up space."
"The Dominators" demonstrated what happens when this trope combines with Ham and Cheese. The Doctor and Jamie get captured by Dominators, who are planning an invasion and running experiments on the natives to find out how intelligent and strong they are. The Doctor gets himself and Jamie to pretend to be stupid in order to put the enemy off their guard ("An unintelligent enemy is far less dangerous then an intelligent one") and Hilarity Ensues. "We don't like the clever ones much! They tell us what to do, you see!"
"Smith and Jones". "I must appear to be human..." "Well you're very welcome to come home and meet the wife, she'd be honoured. We can have cake!"
John Crichton from does this very, very well. Many of his plans rely on the villain of the week dismissing the dumb human as inferior...leading to their ultimate defeat. John is an accomplished physicist and aerospace engineer, who designed the titular Farscape spacecraft to test his own theory, and at one point late in the show he builds his own thermonuclear weapon. From scratch. From memory. Even the rest of the crew underestimates the guy.
Rigel uses this as well, though to a lesser extent. Such as in the first season episode when he was playing a game of chance with a pirate over the location a person the pirate was hunting. After he lost the game and gave the pirate the information, Zhaan was ready to chew him out until he revealed that he'd lost deliberately and given the pirate false information that would lead him wildly off course while still making it look like he'd been trying to win because he knew that the pirate wouldn't leave them alone unless he thought he'd gotten something more valuable. Rigel even complains how difficult it had been to pull off, since compared to him the pirate was a terrible player.
Other characters accuse Constable Benton Fraser of Due South of this, disbelieving he really is that polite, honest, and noble and it's not an act.
In Studio 60 on the Sunset Strip, we discover that Chinese businessman Zhang Tao has been pretending that he can't speak English because "it's fun". Jack Rudolph is not impressed.
Lauren Graham (of Gilmore Girls fame) pulled off this stunt while participating in the second tournament of Celebrity Poker Showdown. Despite being known as a fairly intelligent woman, she would flip her hair and say generally stupid/sarcastic things ("So many numbers!") to throw off her competition. Not only did the other players fall for it, but Lauren went on to the Championship Game of the season, losing only to former co-star Maura Tierney.
While Jayne Cobb is still not the highest caliber shell in the magazine, he acts thicker than he really is: In the pilot, Mal asks him to interrogate a prisoner. When the prisoner tells him a convincing lie (exactly the scenario the prisoner should have been trying to cover up), Jayne sees right through it and accuses the prisoner of not even trying. Then in the episode "Shindig", Badger finds himself groping for a word, and Jayne supplies it: "pretentious." Mal looks at him with disbelief, seeming to wonder himself if Jayne is truly all he seems. And in "The Message," he shows forethought and a slight philosophic interest.
Jayne: [to Shepherd Book] "Me, I see a stiff, one I didn't have to kill myself, I just get the urge to, ya know, do stuff. Work out, run around, maybe get some trim, if there's a willin' woman about...My kinda life don't last long, preacher. I 'spect I'm invested in makin' good sport of it while I can."
Saffron's "real husband" Durran Haymer appears to be fooled by Saffron's story until it becomes very obvious it's false. When she mocks that if he had any sense he would have called the police the moment he saw her, he reveals that he did, and has been stalling her.
Colonel O'Neill on Stargate SG-1, while far from a rocket scientist, usually presented a snarky persona that appeared much dumber than he actually was, often to fool his enemies or simply to annoy his allies. He is an amateur astronomer (when he's not spying on his neighbors through his telescope) and a brilliant strategist/tactician. He is also fluent in Spanish, can beat General Landry in chess, can be diplomatic if he has to, and is smart enough to quickly pick up how to control an alien starship with a little guidance. He is also the only one who can come even close to keep an alien database in his head without dying immediately.
Though Tony's mother Livia on The Sopranos was clearly suffering from some level of senility, it was more than hinted she was playing it up as a cover for her more terrifyingly evil moves.
Colonel John Sheppard of often appears to be goofy and laid back with a predilection of not taking things very seriously. His detractors tend to underestimate him and take his easy-going personality at face value, especially if there are other officers who want his position as Atlantis's military commander. This usually comes back to bite them in the ass when Sheppard goes to prove that his rank and job are rightfully his. He's also extremely intelligent to the extent that the genius Dr McKay was very surprised to hear he could have gotten into MENSA.
In anotherAlternate Universe, Sheppard is a Las Vegas detective after getting kicked out of the Air Force for the Afghanistan mess-up that got him sent to Antarctica in the main universe. He's unshaven, drinks a lot, sleeps around, and a little bit corrupt. Not exactly the sharpest tool in the shed, right? Well, he's pretty good at poker, and he's the only one to figure out where the Wraith is hiding.
Kamen Rider Den-O has a case based on ignorance rather than stupidity: throughout the series, Ryotaro's sister Airi is a sweet, caring big sister whose memory seems to have been altered so she doesn't even remember her fiancé Yuuto Sakurai following his mysterious disappearance. In a later episode, she reveals to Ryotaro (and the teenage version of Yuuto) that she knew all along, and in fact helped create Sakurai's plan to protect the Cosmic Keystone (their future daughter Hana) from the Big Bad. In order to throw the villain off the trail, Airi had to pretend to be innocent and clueless so he never realized she was important until Ryotaro was strong enough to defeat him.
Vila Restal from Blake's 7 is a genius safecracker/pickpocket who spends the majority of his time acting like a cowardly imbecile. While he is a coward, he isn't stupid. There's a reason he's the only protagonist to have survived from the first episode to the last, and the most likely to have survived that Bolivian Army Ending.
In one episode he has an idea for how to repair the engines while in flight, but doesn't want to actually do it because it would be dangerous. He wanders on to the bridge drunk and waving a bottle (entirely in character) and ramblingly describes his plan. Avon and Tarrant go to repair the engines and Vila reassures the rest that you can't get drunk on water.
Most everyone on Burn Notice uses this at some point, being a spy show it is what they do.
On many occasions Michael has had to allow himself to be beaten up as part of his Batman Gambit. At one point he went all out and walked with a lanky stride, had matted and greasy hair, talked about two pitches higher than normal and came complete with an inhaler. For that matter, Michael's normal personality (or generic one when he's working a cover) when he isn't on the job has some elements of being a snarky idiot just to put people at ease.
Sam basically embodies this completely. The initial impression one gets of him is a slightly overweight womanizer who chugs beers and has about as much insight as a sixth grader. While the first part is still completely true, one learns over the course of the series that he's a former Navy SEAL, has numerous contacts across multiple government agencies, and can be just as manipulative and technologically proficient as Michael or Fi.
In an episode of Unhappily Ever After, Tiffany tests the hypothesis that guys prefer stupid girls by playing the part of two people—a smart person with glasses and a Ditz without—for the same guy.
Robert Goren on Law & Order: Criminal Intent's unassuming demeanor and odd mannerisms tend to actually become the frightening point about him when he corners the criminals.
iCarly: In iHire An Idiot, Ashley, the hot intern Freddie picked. She seems to be as dumb as Cort but in the end is revealed to be a very intelligent college student. She only pretended to be stupid as part of Freddie's plan.
Lauren "Am I bovvered" Cooper from The Catherine Tate Show appears to her teachers to be a dim, rude, and dismissive teenage layabout. She is rude and dismissive, but certain sketches have revealed her to be smarter than she appears. She can recite obscure Shakespeare from memory, she has memorized at least part of the periodic table of elements, and can converse in fluent French (and knows her signature catch phrase in Modern English, Elizabethan English, French, and Sign Language). It seems she is pretending to be dim as to not be alienated by her much dimmer friends (while being smart enough to manipulate them).
Andy Pipkin from Little Britain, who onscreen is shown to be a monosyllabic dolt; however in many of the sketches, when Andy says he wants (to do) something he doesn't like, Lou will remind him that he expressed a negative opinion of said thing, some of which make him sound like a philosopher.
Lou (to Andy): But I thought you didn't like tattoos. You once told me they were nothing more than graffiti over God's work.
In Star Trek: Enterprise's "In a Mirror Darkly," Mirror Hoshi pulls it on both the characters and the audience. She seems like nothing more than a willing consort for whoever her captain is at the moment, then at the end she poisons Mirror Archer and declares herself Empress of the Terrans.
The Pakleds in the Star Trek: The Next Generation episode Samaritan Snare are to all indications genuinely some of the most stupid aliens featured in Star Trek, but they are capable of playing it up even more in some areas to appear more harmless than they actually are and manipulate others into doing what they want.
Tony DiNozzo in NCIS comes across as an immature Jerk JockHandsome Lech who spends far more energy goofing off, chasing women, and giving his teammates a hard time than he does actually investigating. When he has to get serious, however, he quickly proves that he is not only a highly skilled investigator, he's also an effective leader, and particularly following the fourth season it becomes clear that most of his immature behavior is a front under which he is a lot smarter and more capable than he lets on.
In one episode of The Mentalist, "Flame Red", the killer is an intelligent young man named Tommy who has been pretending to be mentally retarded for some time, ever since he it got him out of a parking ticket because of the magical words "he didn't know any better." He gets angry after someone he knows is killed for greed, and gets revenge.
Michael Guerin, in early episodes of Roswell, is one step up from being a high school drop-out, playing truant and skipping classes as often as not. In conversation with Maria, his eventual love interest, he claims his favourite book is James Joyce's Ulysses. She scoffs and doesn't believe him...so he recites a very long passage from said book from memory.
Megan in Drake & Josh pretends to be an innocent little girl in front of her parents so they won't believe her brothers when they go to tattle about the not-exactly-harmless pranks she pulls.
Mohinder Suresh of Heroes has many moments of real stupidity, which means his occasional forays into Obfuscating Stupidity can still catch the audience and other characters off-guard. In particular, he successfully pretended to have no idea of Sylar's real identity on an eight-hour roadtrip, even going so far as to invite him into his home and cheerfully offer him tea-which Sylar accepts, silently gloating that he's got Mohinder so completely fooled, only to find himself passed out on the floor five seconds after he actually drinks the "tea."
It's been observed in-show that Seeley Booth's Obfuscating Stupidity serves the dual purposes of making people underestimate him and allowing Brennan to be "the smart one" (granted, she is the smart one, but he lets her think the gap between them is even larger than it is).
Dr. Wyatt: (after Booth makes a wisecrack, appearing ignorant) He does that. He wants to be underestimated.
Bones is guilty of this as well, as several conversations and episodes imply or state outright that she knows perfectly well how smart/dumb Booth is. Later episodes expand the idea as her playing dumb to his particular skills so that he can have his thing.
Like in the earlier films, the '50s Zorro television show had the title character disguise his secret identity with stupidity; however, instead of being a rich fop, his alter ego was a bumbling Zorro sympathizer who supposedly lacked the skills of his idol.
Lothor, who had spent the season being one of the silliest villains in the franchise history, reveals in the finale that he's been playing dumb all season — even his constant losses have a purpose which is to fill the abyss of evil so full with monsters that he could burst it open and unleash all the stored evil on Earth at once.
His nieces Marah and Kapri, Those Two Bad Guys in the series, spend the bulk of the series being portrayed as The Ditz duo, practically a disgrace to be called evil, and failing in every one of their attempts for their uncle's plans. In the third-to-last episode, it's revealed that it had been a façade from day one; they were just as cold, calculating, and intelligent as Lothor himself, and had been biding their time since the start to usurp his power. Then, once their plan seemingly fails and they kill the general that they coerced into working with them, it's revealed that they were in league with The Starscream in their scheme. And then it's revealed that they were Double Agents, and had been Lothor's Co-Dragons in secret since the start. So, not only were they smart enough to fool everyone into thinking they were harmless, they were smart enough to trick everyone aside from their boss, the Big Bad, as to whose side they were actually on.
As a topper, when Lothor returns in "DinoThunder", he attacks the Wind Academy where Marah and Kapri are now training and captures all the other ninjas. The two are able to talk their uncle into taking them back, claiming they can be tough now that they're (semi) ninja trained. Lothor agrees, although warning them that one screw-up and they're done. The two go back into their old ditz routine so well that Lothor is annoyed and ignores them...allowing them to rescue Sensei Kanoi right under their uncle's nose.
Dollhouse: Alpha. The character's reveal gave cold shivers, considering how convincingly the actor had been selling the goofy one-off hilarious fanservice guest star part up to that point in the episode.
During his early TV career, British presenter Louis Theroux ruthlessly exploited a put-on faux-naif persona to lull his subjects into a false sense of security. He also intentionally played stupid until he annoys his subjects to spell out their views, which has been couched beforehand in elaborate euphemisms and doublespeak, in offensively plain language. This was a surprisingly effective interview technique, though it seems that people eventually caught on, and he doesn't do it much now.
A large part of the modus operandi of rookie lawyer protagonist Kuryu Kohei in Hero is to act ditzy and excitable, hiding his brilliant mind.
Used by then rookie defense lawyer Randy Dworkin in Law & Order: After letting all the People's witnesses go unchallenged, he lays out a case basically saying finding his client guilty is questioning the will of God. After the judge shoots this down, he then changes his client's plea to not guilty by reason of mental defect. Normally this wouldn't be allowed at that stage of the trial, but he then reveals his client will have a solid appeal on Sixth Amendment (right to a fair trial) grounds, using his own incompetent defense up to this point as evidence. All to catch the prosecution off guard with the plea change.
Judge: Either you are a brilliant strategist, or you are the biggest jackass to step foot in my courtroom.
Boston Legal's Denny Crane seems like a senile, self absorbed wackjob and he usually is...until he steps into a courtroom. Then Again he is Denny Crane.
Dave, the truck driver who doused Alex in superpower-inducing chemicals in The Secret World of Alex Mack, puts his own safety on the line by foiling the greedy and evil Danielle Atron's schemes to find out Alex's identity. Dave pretends he has no idea who the victim of the accident was, but reveals in the last episode he had discovered the secret in his A Day in the Limelight episode, where he used the skills learned by a rather goofy "How to be a spy" course on tape. He decided to keep quiet in accordance to said course's last lesson: "Whenever you discover a secret, no matter how big it is, it's best to keep it for yourself."
Im the sketch "Masterbrain," Phil Hartman portrayed then-president Ronald Reagan as a doddering-yet-genial goofball (the image that was and still is popular among his detractors) when presenting himself before the press or the public. Behind closed doors however, Hartman's Reagan revealed himself to be a Machiavellian manipulator in full command of the issues of the day (while everyone else in his Cabinet is struggling to keep up).
Another sketch had Marilyn Monroe acting the Dumb Blonde she was known for in public, while behind closed doors showed her as the brains behind JFK's presidency while he was a moron who only acted smart in public. The first part of this premise actually has some basis in truth, as Norma Jean Baker/Marilyn Monroe was well read and her breathy voice and deer-in-headlights expression were something of a put-on.
Sean Connery on the Celebrity Jeopardy! sketches. Originally, he was just as stupid as all the other celebrities, but when the focus shifted to his adversarial relationship with Alex Trebek, it became obvious that he was just faking it to get his goat. (Sometimes, he even admits as much himself.)
On one episode of The Steve Harvey Show, Lydia (with encouragement from Romeo and Bullethead) dumbs herself down when her boyfriend Arthur starts to lose interest in her because she aced a chemistry exam.
Whitley plays with this trope on an episode of A Different World. When she and Dwayne are partners in a quiz bowl, Dwayne tells her that their kiss on a previous episode meant nothing and that he was in love in Kinu. He also alludes that Whitley is not very smart and that she is spoiled. Whitley becomes angry and sabotages the second half of the game by giving wrong or flat out stupid answers, and at one point not giving any answers. She even files her nails in one scene. Of course Hillman is eliminated from the tournament.
Carla's brother from Scrubs pretends to not have learned English so as to play with Turk's head, whom he's had a rivalry with ever since Turk mistook him for a valet at Carla's mother's funeral. This is only revealed to Carla when Turk provokes him enough for him to yell out in English, "That's it, you sonofabitch!".
Mr. Eldridge on Remember WENN usually comes off as The Ditz, but in the episode where he wins the lottery, he manages to use his winnings to buy a controlling interest in the station and save his own job.
The title character of Merlin does have his moments of clumsiness and cluelessness, but sometimes he may use it to his advantage.
In one episode of All in the Family, Mike is spouting off about how clueless Archie is when Edith tells him, "If you were really smarter than Archie, you would be smart enough not to let him see that you're smarter than him." (Which offers a tantalizing hint that Edith's ditzy exterior might be at least a partial application of this trope in its own right.)
In the early seasons, Lionel Jefferson would have fun with Archie by adopting a stereotypically "black" version of this around him.
Eliot from Leverage. Ostensibly just the dumb heavy, he's actually quite intelligent and enjoys activities such as gourmet cooking. In the "Zanzibar Marketplace Job", he ends up running the entire con-slash-rescue mission.
Maggie: You know, people underestimate you, Eliot Nate: That's kind of the point.
Caprica: Serge Graystone, the Robot Maid about the size of a trashcan, has a twitterfeed that suggests he knows more than he's letting on - and that he's not simply very well programmed to fake sapience.
"I may seem dumb, but that's just to get your mother to not ask me to do stuff!"
In Being Human (UK), George does menial labor in a hospital and is later revealed to have "a brain the size of a planet". Of course, there's the problem that he turns into a wolf every month, so it makes sense that he'd want a low-profile job with access to a room to lock himself up in. The first season finale gives another example, when he appears to be scared enough of an evil vampire to abandon Mitchell. It turns out that he'd cleverly set up a trap in which the evil vampire would find himself locked in an isolated room with George...just as the full moon comes up.
In Sherlock, Molly's dumb, Camp Gay boyfriend "Jim from IT" is revealed to be James Moriarty, the diabolicalcriminal mastermind who outwits both Sherlock and his incredibly influential (and intelligent) older brother Mycroft, and very nearly succeeds in destroying Sherlock's life completely.
Nicaragua sees Jud Birza, better known as "Fabio", using this as his strategy the whole time. Almost the whole game, Fabio had been playing the male dumb blonde stereotype. At a certain point in the game, when there were 6 people left, there is a four person alliance (Sash, Chase and Holly) against a three person "alliance" (Fabio and Dan and Jane, who weren't in an alliance, but didn't have much to go on). In an incredibly subtle and clever move after winning immunity, Fabio put an idea in the heads of Sash and Chase that Jane was a bigger threat than Dan, their original planned target, and they took the bait. Fabio then proceeded to slink back into the shadows as Sash and Chase made the idea entirely their own and Jane found out about it. By this point, they had completely forgotten that the idea was Fabio's. Cut to Jane putting out the fire angrily before tribal council... and Fabio grinning maniacally when nobody could see him.
The best thing about this? He started playing dumb because he looked the part.
This has actually proven to be an excellent strategy in Survivor. Pretend to be stupid so that people drag you along and underestimate you, and then start turning everyone around and against each other, as well as playing the friendly person so that you can win the personal vote, too. This worked for Vecepia who flew under the radar in Marquesas and got votes because Neleh was The Load. It also worked for Natalie in Samoa, who noticed that Russell Hantz was eliminating everyone who wasn't Too Dumb to Live (Russell said that Marisa and Betsy were the first to go because he felt they could have beaten him) and aligned herself with Russell so that he would drag her along. Even further, she pretended to be The Load along with Mick so that Russell would never pull a You Have Outlived Your Usefulness on her like he did on Shambo and Jaison. Then when the Jury questioned everyone, she admitted she was the least deserving while Russell slammed the other two and bragged "Aren't I awesome?". The result? She won seven votes. This goes to show you that if you align yourself into the middle-or-low man on the totem pole, you are perhaps in the best spot.
Brenda Johnson in The Closer is a sweet, scatterbrained Southern eccentric - until her suspect slips up and gives her what she wants. At which point said suspect is painfully reminded that Brenda Leigh Johnson is a CIA-trained interrogator.
Most of the main characters in Misfits could fit, as all of them are smarter than they let on. The best example is possibly Simon, who at first appears to just be a creepy weirdo, but soon becomes The Smart Guy, organizing the group to get away with murder. Twice.
Papa Lazarou from The League of Gentlemen uses this as part of his stage act to confuse his volunteers (mostly women) to hand over their jewelry.
Shawn Spencer of Psych got a perfect score on the California detective's exam at age fifteen, identified T-rex bite marks from a time he climbed on a museum exhibit several years previous, and once fired a gun from the hood of a moving vehicle, between the slats of the grill of another moving vehicle, after being shot in the shoulder. He generally acts as though he knows absolutely nothing about anything apart from pop culture and detective work (and the latter only with Gus and his father). This is an odd example, however, because he does this not to get people to underestimate him, but because he needs to believe he's a somewhat normal human being.
Declan Rand ("Shawn 2.0"): "You're highly intelligent, but you're shameful of that fact so you play it down with the use of inappropriate behavior. And you live in fear of showing weakness so you hide behind a constant barrage of jokes and sarcasm."
Sergeant Schultz on Hogan's Heroes repeatedly states, "I know NOTHING!" but in reality he's probably on to just about everything that goes on in Stalag 13. At one point, he stops Hogan's crew from pushing him too far on a deal: "Sometimes I have to work for OUR side!" This was reportedly a conscious decision of actor John Banner's, to portray a decent human being who just happened to be in the German army and to subvert the ironic pattern of typecasting as a Nazi that Banner (who was Jewish and escaped Austria one step ahead of the Nazi occupation) experienced in his career.
In-series, it's also revealed that Schultz not only managed to get himself promoted to Hauptfeldtwebel (that's Senior Master Sergeant, or E-8 equivalent), he also ran a highly successful toy company in the interbellum years, and only re-enlisted because he had no better options when his factory was seized for the war effort.
Tom Ballard from Waiting for God has been doing this so long he has trouble working out when he's doing it and when he's just being mentally lazy, leading to an unusual case of Becoming the Mask.
Lt. Commander Bud Roberts on JAG once used this as a means to get a serial killer who felt it was his duty to remove the "waste" from the Navy to confess in court.
Sarah especially seems fond of adopting the role of a ditzy girl who has drunkenly wandered into whatever situation she's in by accident, which allows her to walk out of places she's found herself trapped in and serves as a good excuse to walk into places where she would otherwise be immediately shot dead or at least regarded with suspicion.
Chuck himself falls back on this at times, relying on his nerdiness and $12-an-hour Buy More job to make the bad guys assume that there's no way he's a spy.
Alexi Volkoff is a villainous example, at least in his first appearance. His first episode characterizes him as a mostly incompetent, bumbling agent. Then The Reveal hits and he turns out to be one of the most dangerous characters in the show, if a bit eccentric.
Castle: At a shooting range, Castle first appears to be a terrible shot. Until Beckett promises to give him the files he wants, if he's able to hit the ten ring. He sinks the next three bullets exactly there.
Castle: You're a very good teacher.
Castle in general. In the first 4 seasons, he spends most of his time cowering behind Ryan and Esposito in takedowns, goofing off, flirting with/annoying Beckett, and coming up with CIA conspiracy theories. The fact that he's taken down a trained assassin while Beckett got nerfed; shows excellent marksmanship; has saved Beckett's ass at least as many times as she has his; and got further in the span of about 2 years on Beckett's mother's case than the police or Beckett herself managed to in 10, says he's not quite as hapless or physically ineffectual as he lets on. By the second half of the 5th season, the show had largely given up this angle, but it makes an appearance now and again Depending on the Writer.
Skyler's old boss Ted becomes noticed by the IRS CID, due to massive tax fraud that she helped cover up. She fixes it by pretending to be a Dumb Blonde secretary who only got the job by sleeping with Ted, and hopelessly screwed up the company's accounts by stupidity rather than intentional fraud. Plus, she gets revenge on Ted for getting her involved by not telling him about the plan beforehand, and making him sweat through part of the audit alone before showing up late as part of her character.
Saul Goodman is an extremely talented lawyer and business associate, but has no problem letting Albuquerque think he's sleazy and incompetent.
In the "Village Idiot" sketch from Monty Python's Flying Circus, John Cleese's idiot is portrayed this way. When nobody else from their village (except possibly other idiots) is around, he'll discourse learnedly to the camera on the "vital psychosocial role" played by idiocy; as soon as someone walks by, he slips back into character and starts babbling nonsense and comedically falling off walls.
Del Boy from Only Fools and Horses sometimes does this when he wants to manipulate someone else into having an idea that he's had, and thinking it was their idea in the first place. For example, in one of the later episodes of the show (after the Trotters have become multi-millionaires and lost it all), Del has been declared bankrupt and banned from being the manager of Trotters Independant Traders. He realises that someone else - namely, Rodney - could manage Trotters Independant Traders and hire him to work for them. Instead of just asking Rodney if he'll be the new manager, Del proceeds to manipulate Rodney into coming up with the idea of managing the company himself. He even pretends to not understand what Rodney is suggesting and that he needs it explaining to him again, presumably to strengthen the illusion that it was Rodney's idea.
The Inspector Lynley Mysteries: Detective Sergeant Barbara Havers is tiny and adorable, with a pair of enormous green eyes and an accent that thoroughly betrays her working-class origins. She uses all of these to devastating effect, playing the "silly, uneducated little girl" card right up until she gets what she needs from her suspect - at which point she reveals that she is more than a match for her Oxford-educated partner in terms of her detective skills. She might not be an Oxford grad, but she is a frighteningly competent detective with several decades of street smarts who is much, much smarter than anyone would guess.
While Gomer Pyle, U.S.M.C. was built on the idea that the titular character was a dumb-as-bricks hick out of place, more than once he was shown to have exceptional wartime skills, such as him and the Sergeant being the only team in the survival exercise to gain weight, or dismantling a wooden bridge that was to be simulated as detonated so he had ropes to build snare traps with.
Detective Lester Freamon is the master of this on the first season of The Wire, convincing all around him that he's a "hump" who, in the words of his new commanding officer, "couldn't find his own gun." When he finally does reveal his inner Chessmaster to his colleagues, they literally stare at him slack-jawed. He still keeps up the facade to the bosses, though, which pays big dividends in the season finale: despite being the prime mover behind the decision to investigate the politicians who took money from Big Bad Avon Barksdale, he's the only member of squad who ends up being rewarded by the bosses for his work on the detail, receiving a transfer to the Homicide Squad while most of the other detectives were being Reassigned to Antarctica.
Alan Davies, the permanent panelist slash resident Butt Monkey from the Panel GameQI. He admitted that the show requires someone to do the dirty job of feigning ignorance from time to time, and he usually steps up to do it.
On White Collar, Peter does this from time to time. For example, in "All In" he deliberately plays the bumbling agent, lets a bunch Mei Lin's co-workers appear to confound him by speaking only Chinese at him, and then walks away — and reveals to Neal and Lauren that he was recording the whole thing for later translation.
Ryan O'Reilly was a Chessmaster and used Obfuscating Stupidity to make sure his targets never suspected they were being played.
When Chris Keller plays chess with Verne Schillinger after having apparently been taught by Tobias Beecher, it's implied that he already knew how to play and was pretending not to in order to bond with Beecher.
William Giles suffered from dementia, but pretended he was further gone that he really was. While on death row and asked to pick his method of execution, Giles said that he would like to be stoned to death, knowing that he would likely die of old age while his execution was stalled by lawsuits from human rights organisations.
The title character of I, Claudius is a famous example.
The bad guys in the first season 2 episode of Las Vegas use this to infiltrate the Montecito surveillance room.
In the Will and Grace episode "May Divorce Be with You", Karen is trying to divorce her husband, Stan. Since Will is Stan's lawyer, Karen has to find one of her own. She ends up with a young, bumbling kid named Jason "J.T." Towne, a.k.a. Soupy (played by Macaulay Culkin). J.T. presents himself as completely incompetent and childish. During a discussion with Will, J.T. has a panic attack about the thousands of dollars Karen has at stake (which Will reminds him is actually millions). Feeling sorry for him, Will reveals some extremely helpful advice to him. Soon after, J.T. reveals it was all an act in order to get the opposing lawyer to do all the work for him.
In the Criminal Minds episode "Proof" Ben Bradstone is genuinely dumb. He is not as dumb as he acts however, and intentionally plays this to lure victims in.
Clu from So Weird seems like nothing more than a beach bum wanna-be, but he often notices the weird things happening almost before Fi does. In season 2, he was accepted into UC Santa Cruz, where the minimum required GPA for out-of-state students is 3.40.