Samantha "Sam" Sparks in Cloudy with a Chance of Meatballs was secretly a nerd in her school days but hides her brains behind the facade of the pretty, ditzy weathergirl for fear of being stigmatised. It adds another layer to the character that she is voiced by Anna Faris who is more well known for playing genuinely ditzy characters.
Bill: (on being incompetent fathers who use incompetence to get out of chores) You see, we are dumb, but we're not so dumb.
Captain Jack Sparrow, Pirates of the Caribbean. You might think he's quite a harmless fellow when you lock him in chains, but if you start a 10 minute long conversation with him, he'll probably talk you into opening his chains, steal your pistol and threaten you with it, make a break for it and steal the ship you should be guarding before you can even realize what the hell just happened. And when you reach out for your pocket to get the whistle and call the guards you'll notice that your wallet is also missing...the quotes below give great examples:
Norrington: "No additional shot nor powder, a compass that doesn't point north...(looks at Jack's sword) And I half expected it to be made of wood. You are without doubt the worst pirate I've ever heard of."
Captain Jack Sparrow: "But you have heard of me." —>(Shortly after, when their ship is stolen by two people, including Jack)
Lt. Groves: "That's got to be the best pirate I've ever seen."
Prior to joining Starfleet, it's even stated by Pike that Jim's the only "genius-level repeat offender in the Midwest."
If you watch The Wizard of Oz closely, you will notice that the "brainless" Scarecrow is the one who has all the ideas, comes up with the plan to sneak into the witch's castle, and even (at the climax) manages to think quickly enough to use the Tin Man's ax to drop a chandelier on the witch's soldiers. This is even more pronounced in the original book, where he comes up with clever solutions to nearly every obstacle and monster that the Witch of the West and Oz itself throws at them. In both versions, the Scarecrow's dumb act has even fooled himself. He associates intelligence solely with being a "great brain" or "deep thinker". Since he's not one of those by any means, he fails to recognize his planning, quick thinking, and resourcefulness are also signs of intelligence.
He's so in denial of his own intelligence that he resorts to Self-Deprecation. When Dorothy points out to him that he came up with a clever plan to save her, his only response is to say that if he had a brain, he would have thought of it sooner.
Of course, that was rather the point - the Cowardly Lion and the Tin Man also always had the traits (bravery and heart) that they wanted the Wizard to grant them, which is why he gave each of them nothing more than an item to act as confidence builders.
In the musical version of Wicked, it is implied that the Scarecrow, actually a transformed Winkie prince, knew he was the most intelligent of the group, but faked stupidity so that he could join up with Dorothy and return to Elphaba.
Miss Gulch: I'm all the lame for the bite on my leg.
Uncle Henry: You mean she [Dorothy] bit you?
Miss Gulch: No, her dog.
Uncle Henry: Oh, she bit her dog! Whacks the gate closed onto Miss Gulch.
Miss Gulch: No.
Zatoichi, in the series of movies of the same name, posed as a harmless blind masseur (he really was blind) until it came time for either intimidation or the mass slaughter of the Evil Minions.
In at least one movie he actually can see, but pretends to be blind in order to gain a blind person's insight. Seriously. It makes more sense when he says it.
In the 2006 Pink Panther remake, it is revealed that Inspector Clouseau could speak Chinese all along and figured out who the killer was when the Chinese woman named "Yu" was brought in for questioning. His rant about actually knowing Chinese turned out to be true.
In Revenge of the Pink Panther, a character expresses the (erroneous) belief that Clouseau's blundering is actually a version of this.
Budd from Kill Bill is depicted to be a drunken redneck who works as a bouncer for a titty bar. Nonetheless he has shown a philosophical side and is the only Viper to get an advantage on Beatrix Kiddo, by faking unawareness that she's hiding around his trailer and blasting her with rock salt when she busts through his door. In fact, Budd is the only of the vipers not to be killed by The Bride (well, Elle is not actually shown dead, but is at least very thoroughly screwed).
With the recent announcement of Kill Bill Volume III, Elle Driver is confirmed to be alive.
He's also the only one with whom she seemed to exercise caution. With everyone else, Bill included, the barges in and starts swinging, kicking, and slicing. When the woman who takes on an entire army of sword-wielding Yakuza by herself skulks around a guy's trailer and takes the time to make sure he's sitting down before making her move, and he still gets the best of her...you know he's a badass.
Silent Bob, from Kevin Smith's View Askewniverse appears to be little more then Jay's silent companion, only less exuberant and dumber. It turns out that he's quite knowledgeable. Silent Bob may seem smarter than Jay because his friend is very loud.
Yoda in The Empire Strikes Back is introduced this way, pretending to be an irritating little creature who steals food and squabbles with R2 before finally revealing himself to be the ancient and powerful Jedi master. It's all a Secret Test of Luke's patience.
Jackie Chan tends to play these kinds of characters, particularly in Rush Hour, when he allows Chris Tucker's character, Carter, to believe he doesn't speak English. Carter gets him back, though, when he reveals that he can speak Chinese just fine. The Screwups Reel makes it quite clear, however, that Tucker himself can't, and since he doesn't speak Chinese in the second movie either, it's possible that Carter only knows the basics of the language or learned just enough for the prank.
In both the 1940 and 1920 versions of The Mark of Zorro, Diego de la Vega plays the featherbrained fop to lull the corrupt Governor Quintero, while, as the masked swashbuckler Zorro, trying to frighten Quintero out of the country.
And the 1974 made for TV movie with the same title. Frank Langella's portrayal of foppish Diego de la Vega is the greatest bit of the film.
The janitor in the horror film Disturbing Behavior pretends to be mentally handicapped because it causes people around him to drop their guard and become "interesting."
In The Usual Suspects, Verbal Kint, pretending to be a weak-willed and crippled sap who was taken advantage of by Dean Keaton, rather than the diabolical crimelord he is.
In Superman III, villain Ross Webster's assistant and girlfriend Lorelei acts the part of the Dumb Blonde but is secretly highly intelligent (when alone she reads Immanuel Kant's Critique of Pure Reason and disputes one of Kant's arguments). As a result she's able to outwit Ross and his sister.
In Horse Feathers, Connie plays up her football ignorance to try to get Professor Wagstaff to share his secret signals. She overdoes her childlike ditziness, so either he sees right through the ruse or he thinks she needs to snap out of it.
Roger Rabbit spells it out in Who Framed Roger Rabbit, and implies that obfuscating stupidity is second nature to a toon:
Roger: (to Judge Doom and the Weasels) We toons may act idiotic, but we're not stupid!
Given that he never actually does anything useful in the movie, it could just be that he's full of hot air on that one, which would also just make him stupid rather than using Obfuscating Stupidity.
The toons in general seem to vary in intelligence. It could be that Roger is just a stupid one.
Subverted in The Court Jester: some of the villains, believing the title character is either an assassin they hired or a Robin Hood-esque resistance leader, believe he's doing it. In fact, he's actually a carnival performer who really is that twitchy and bumbling.
Sir Roger Moore's James Bond uses this several times.
Dewey: "How do you know that my dimwitted inexperience isn't merely a subtle form of manipulation, used to lower people's expectations, thereby enhancing my ability to effectively maneuver within any given situation?"
(a brief pause, then Gale starts laughing)
In Mr. Baseball, Tom Selleck plays an American baseball player who signs up with a Japanese team. The team's manager appears not to know English through most of the film, causing grief for Selleck's character when he finally finds out.
True Lies does this beautifully several times, though it borders on ridiculous when Harry calls the obvious nuclear warhead an espresso machine...of course at that point it's clear he was being sarcastic to annoy the terrorists.
Summer of '42 has its underage protagonist employ some of this to obtain some condoms from a pharmacist. When asked what he wants them for, he says they're actually for his (nonexistent) older brother and that he assumes they're for filling with water and dropping on people from a roof.
In Dumb and Dumber, the kidnappers only think Harry and Lloyd are doing this.
In Maverick, Bret Maverick pulls this off effectively in an early scene where he invites himself into a poker game, saying, "I promise that I'll do nothing but lose for at least an hour." For the first hour of so he indeed loses every hand. From second hour of the game on, Maverick winds up cleaning out the rest of the table. When one fellow gambler accuses him of cheating, Maverick explains, "What do you think I was doing for the first hour? I was learning your tells." This sets the tone for the rest of the movie, in which Maverick appears to be bumbling his way through the West, but in reality is as shrewd and sly as anyone else in the movie.
In a more specific example: in The Dark Knight, Joker causes chaos by threatening to blow a hospital unless the people kill a man who claims to know Batman's identity. When someone tries to ram the police vehicle carrying the man, Bruce pulls a vehicular Taking the Bullet. Afterwards he acts like it was an accident, but still shoots a knowing glance towards the man whose life he saved.
Gordon: That was a very brave thing you did.
Bruce: Trying to catch the light?
Gordon: So you weren't protecting the van?
Bruce: Why? Who's in it?
Bruce: (after shooting the knowing glance) Do you think I should go to a hospital?
Gordon: You don't watch a whole lot of news, do you, Mr. Wayne?
Interestingly, the third film reveals that Gordon had no idea that Bruce was Batman (that's right, Gotham's top cop is the last named character to figure out who Batman is).
Ofcourse he pointedly refuses to even try to find out who Batman is.
The Hallelujah Trail: Walks Stooped Over, the Sioux chief who does all of their negotiating in badly translated sign language, is revealed at the end as speaking perfectly good English.
Both Mr. Green and Colonel Mustard both embody this throughout most of the film Clue.
Although, depending on which ending you get, it's completely genuine for Green.
In The Avengers, Natasha Romanoff uses a variation of this combined with a Wounded Gazelle Gambit as an interrogation technique. She lets the interrogat-ee think that they've outwitted her and plays the part of the helpless prisoner, then waits for them to shoot their big mouth off while they're gloating about how she failed to live up to her reputation. Then she breaks out and kicks the shit out of them.
David "Mo" Rutherford, in The Stuff. He portrays himself as a quite inept and dimwitted lowlife, introducing himself in a particularly obnoxious manner. The trope quite soon lampshaded when one character, impressed with Mo's ingenuity in placing a small microphone into the pockets of everyone in present at the meeting on the night before, comments "you're not as dumb as you appear to be". To which he responds "Nobody is as dumb as I appear to be".
Mr. Shhh (played by Steve Buscemi) in Things to Do in Denver When You're Dead. He's introduced with narration showing up in a slightly disheveled suit and slumped shoulders. He proceeds to kill a couple of muggers without breaking a sweat.
In My Fellow Americans, former Presidents Kramer and Douglas run for their lives, trying to discover the criminal genius who framed them for a bribery scandal. Turns out it was the supposed fool of a Vice President, Ted Matthews, who arranged to remove both the former Presidents and his own boss with a single scandal.
In Iron Man 2 Vanko shows he's both fluent and eloquent in his early face-to-face confrontation with Tony Stark, but speaks to Hammer in broken, barely intelligible English just to dick with the guy. Later on he acts only barely competent at engineering and technology, convincing Hammer that the best he can do with the Hammer suits is to make drones and then later on that the best he can make the drones do is "salute." Hammer naturally underestimates him as a result.
Big time in Now You See Me. Rhodes is always one step behind the Four Horseman and is always outwitted by them. He's actually the Fifth Horsemen, deliberately pretending to fall for all their tricks to make sure the authorities are always one step behind the Horsemen.
In Django Unchained, Stephen plays the role of a submissive house slave in public. In private however, he's actually smarter than his master and serves as Candie's personal adviser.