Nao Kanzaki from Liar Game, to an annoying extent. While at first she was thrown into a world of lies and tricks, despite being "honest," she doesn't seem to understand the point of the Liar Game. First tricked into giving 100 million yen to a "trustworthy" teacher, then almost screwed Chessmaster Akiyama's plan by making a part of the team the most unreliable person, Mr. X, in the round and then trusting said unreliable person, after she had learned of Mr. X's true nature. Between her mistakes, she shows some bits of understanding her situation, but she forgets them when it's important. If it weren't for Akiyama, she'd be screwed by now.
In the drama series this is very much true. However, in the manga series it becomes less as time passes. In the current round, she managed to see through Yokoya's so-called psychic powers by putting something other than money in her suitcase (a teddybear) which he obviously couldn't know about. Also, when the team threatens to fall apart she convinces everyone to keep working together.
Putting Mr. X part of their team wasn't exactly her fault, since at that time, they didn't know who Mr. X was. And as noted above, it happens much less later on in the manga, With Nao actually pulling off some of her own tricks.
Atsushi Arisawa from Kusatta Kyoushi No Houteishiki is a Genius Ditz who consistently gets some of the best grades in the country, was invited to an elite highschool in Tokyo without an entrance exam and who sometimes has enough insight on other people's relationships to identify core problems and underlying psychologies that no one else sees. Unfortunately he has no idea that he's actually that smart, goes to a highschool for delinquents, doesn't see why he was invited to an elite highschool in the first place, completely misses the most obvious aspects of most people's relationships, and has the surface mentality and naivety of a small child on a sugar high! Yes, this is very frustrating!
Max from Sam & Max. At times, he's making lewd sexual come-ons and innuendo about sadomasochism, masturbation and drug use; at other times, he's a childlike Chaste Hero who doesn't even understand the biological differences between men and women. It's lampshaded in his Image Song in the Telltale games, which flat-out calls him "a sandwich of naivety and cynicism."
This tendency gets lampshaded every so often, when Max will ask when he can find nubile young women, and Sam will say "Max, you don't even like girls." Also, as the trope description indicates, Sam & Max has quite a bit of Negative Continuity going, to the point that Sam and Max will sometimes mix up each other's names.
Katie Power, the youngest member of the Power Pack, sometimes seems to be written younger. In the All-Ages version, this is mostly for the sake of humor.
Mortadelo y Filemˇn: Filemˇn can either be the Straight Man with much more common sense than Mortadelo or just as big as an idiot as him (usually when the boss is present and both of them do something to anger him).
Besides his famous Sherlock Holmes stories, Arthur Conan Doyle wrote a series of humorous adventure stories about Large Ham Napoleonic soldier Brigadier Gerard. While in some stories, Gerard is rather clever and appears competent despite his inflated opinion of himself. On the other hand, he is a dolt in others and very much an Unreliable Narrator.
Vala Mal Doran's knowledge of popular culture in Stargate SG-1 is inconsistent in the first few episodes of season 10.
Summer Roberts from The O.C. swings wildly back and forth between being a clueless, uniformed airhead and a savvy student who aced her SAT scores and got into Brown.
Summer: Kidding, I'm not stupid, just shallow.
The 'bots on MST 3 K do this on purpose to mess with Joel and Mike.
Adrian Monk is, for example, still convinced that his college roommate's girlfriend was helping him organize whenever there was a tie on the door, or similar things whenever it's funny, but always becomes an expert on human behavior just in time for The Summation.
In 3rd Rock From The Sun, the aliens' knowledge of Earth. One particular example is a second-season episode in which it's mentioned that Sally is going bowling. A fourth-season episode had her going bowling for the first time and not even knowing what it was. For the first two seasons, the Solomons were terrified of Jell-O (thinking it was a dangerous alien entity) yet in a first-season episode Sally mentions making "Jell-O with cheese." Dick is sometimes supremely sarcastic, but at other times he doesn't seem to be able to recognize sarcasm and interprets everything literally.
A particularly severe example is the episode when, very late in the show's run, and after hundreds of scenes in restaurants, the aliens finally figure out what "tipping" is.
This is worked into the plot, however, as Mary is horrified to find out that Dick hasn't been tipping the waiters when they've been going out. Suddenly all the rudeness on their return visits makes sense to her.
Happens all the time on House, with characters who have known the eponymous Jerkass doctor for years expressing varying levels of shock every time he does something insane or unethical, to the point that the one time somebody reacted some other way instead, it was a symptom of a horrible brain-eating parasite. These same characters, however, will at other times act familiar with his behavior patterns, usually for a gag. All of the above also applies to his rudeness toward patients and tendency to pull the correct diagnosis seemingly out of nowhere just when things look bleakest.
And it even happens to House himself. He's deeply cynical about the state of things most of the time but when it comes to dealing with a hostage-taker with a gun, Vogler, Tritter, Stacy and sometimes problems with Wilson and Cuddy, he's naive and mainly likes to keep his head stuck in denial-land.
Played for laughs, a lot, with Liz Lemon on 30 Rock. She's the head writer for a comedy show that mocks pop culture in one scene, and then she'll pull out archaic slang like "discotheque" un-ironically in the next.
Nearly everybody on Heroes displays this at some point. Peter and Mohinder are the most frequent, but even general BadAsses like Noah Bennet and Sylar have conveniently forgotten things they learned or should have figured out several episodes earlier.
Bones demonstrates this perfectly. She is baffled by standard cultural references that everybody else (Booth, Cam, etc) understands while constantly not understanding human motivations, yet she is a best selling author with believable characters in her books who supposedly comprehend those same cultural references and have those same motivations that she doesn't understand in her real life.
New revelations reveal a new explanation in this instance. Someone else (Angela) writes the parts of her books that aren't technical.
It's not that she doesn't understand human motivations, it's that she lacks the understanding of the logic behind it. For example, she understands that Booth believes in God, yet does not see why he does. You can write something without agreeing or understanding it. Ask any high school student.
It also helps that many of her characters are based on people around her, making it a simple copy/paste sort of thing.
She actually seems to be fluent in several cultures, just not her own.
This is actually a fairly realistic portrayal of what many (but not all) people with Asperger's syndrome (which Bones has) are like to varying degrees.
Used to a frequently irritating degree for cheap jokes on NCIS, usually at the expense of either Tony or McGee. Gibbs knows about technology (or doesn't), Ziva understands idioms and pop culture (or doesn't)... etc.
The Doctor, as a character trait. Even the Tenth Doctor, considerably more sexually/romantically inclined than most of them, had moments of being Oblivious to Love (sharing the bed with Martha in "The Shakespeare Code," being offered a phone number in "Partners in Crime").
His naivitÚ to Earth pop-culture in general, as well. In his earlier incarnations, he was definitely a Fish out of Water, so that's fine. But now he's on his Eleventh incarnation, fanboys over famous human artists, and has made references to everything from Ghostbusters to The Lion King. Which makes you wonder how he forgot what football/soccer was.
In particular given he seemed to mistake it for cricket, which he has played before quite well and displayed a knowledge of in his fifth incarnation.
Perhaps justified in that, especially by his Eleventh incarnation, the Doctor has a fantastic quantity of knowledge about the universe and just like is often the case with even normal humans, the more you know in general the harder it is to at least immediately recall specific pieces of information.
The Eleventh Doctor's sexuality is all over the place in particular. When Amy and Rory kiss in front of him, he acts grossed out, like a child, and when River kisses him through the bars of her prison he flails his arms around and looks intensely confused. But there's also Something Else Also Rises jokes in "Pond Life" and "The Crimson Horror", he non-consensually snogs both Tasha Lem and Jenny (when Jenny slaps him for it, he just laughs and says that her slap felt good; and when Tasha protests, he flirts and it's clear she secretly liked it), and then there's the parts in "Time of the Doctor" where in the same scene he has forgotten that nudity is socially inappropriate, and slaps Clara on the bum in front of her parents as part of their Undercover as Lovers scheme.
The titular character from Sherlock. In one episode he's so oblivious John has to explain to him what a date is, while in another he is able to successfully manipulate another character by appealing to her romantic feelings for him. And then in another episode he correctly deduces that this same character is in love with someone, but is very surprised to find out that it's him.
It's heavily implied that he's being deliberately obtuse about the date in question. And understanding sexual attraction is a bit different than understanding unrequited affection.
Frasier: Daphne's knowledge of Niles' feelings. Sometimes it seems obvious that she must know (or at least suspect), while other times the plot only makes sense if she hasn't got the slightest idea (for example, overhearing that "Dr. Crane" is in love with her and assuming it's Frasier).
After Booker T became "King Booker" in June of 2006, he not only started acting like an Upper-Class Twit but apparently underwent a personality disorder of some sort that caused him to forget everything he had experienced as Booker T. When he saw Edge backstage, for instance, he would demand to know who he was - despite having bitterly feuded with Edge just a few years earlier. At other times, however, the old Booker T persona would reemerge and he'd once again speak in his stereotypical "Harlem" accent.
Also might apply to the many wrestlers who underestimate Rey Mysterio Jrafter he has already beaten them! John Bradshaw Layfield definitely counts (and, in fact, this trope ultimately cost him his entire career), but, shockingly, also perennial babyfaceShawn Michaels, who once taunted Mysterio to go back to the locker room because "Hornswoggle isn't out here yet!"
Pretty much the case whenever a gimmick match is set up (typically a cage match, or some other way to "isolate" someone) that will "prevent them from cheating." You just know that means that they'll still get their hands on a weapon, or someone will still interfere from OUTSIDE the match, or even a no-disqualification match (which should let the good guy be on equal footing by being "allowed to cheat") just lets the bad guy cheat even WORSE. Usually because that's the exact same situation they were in a month ago.
Referees in professional wrestling rely very heavily on this trope, usually being completely oblivious to cheating and easily fooled/distracted by heels who are well known for underhanded tactics, but always managing to be far more observant and alert when the situation calls for it.
Phoenix Wright of Ace Attorney. In case 5 of Ace Attorney (the first game). He shows himself to be a competent and clever lawyer, almost to Badass level. In most other cases (i.e. when he has a sidekick, excluding 1-4 & 2-4), he shifts from The Ditz to nearly Too Dumb to Live. Whether these are extreme cases of Obfuscating Stupidity or not is debatable, but the fact remains that while he cleverly outwitted the Chessmaster Chief of Police (Damon Gant), only to revert back to dumbass rookie status by the next game.
Natalia from Tales of the Abyss. She's a bit spoiled, but otherwise elegant and conscious of her responsibilities as a princess. This doesn't stop her from believing every single tall tale that Anise tells her, including the ones about Van's eyebrow powers. Yeah. Eyebrow beams.
Estelle from Tales of Vesperia is the same. She knows that combat is necessary and feels A-Cup Angst towards Judith, on other things however, and well...
Celia of The Order of the Stick gives this vibe. The first time we see her, despite the protections of the castle, she's uncomfortable at the growing dangers of her job and was a sentence away from ditching it. Then she's convinced by Nale that she's 1000 years into the future (although that actually was plausible in context). Then she's a Savvy Lawyer. Then she fails to realize that humans don't have inherent lightning powers. Then she realizes that murdering their way through the hobgoblins would only alert them to the Resistance's presence. Now she's down again, as she's now incapable of recognizing the dangers of a really bad city, can't seem to grasp the bad intentions of the Golem maker, and just lost Roy's bones.
Maybe she's just really bad at Sense Motive, and thus easily fooled by Bluff checks. She's pretty smart otherwise, with the minor exception that she's not real clear on the differences between humans and sprites.
Fish out of Water is the explanation that is invoked with her. It works better than you might think; she looks extra-foolish from our point of view since we know what she's getting wrong, but that's from our point of view. Of course, this is not to say it doesn't get done in an exaggerated way as well.
A lot of that comes down to Genre Savvy being a class feature for bards in this 'verse.
Joey from A Game of Fools is beyond a doubt the most sexually experienced of the main characters and his sheer ease at picking up women is frequently commented on (and shown). He also does stuff like this.
That said, when she's naive she's REALLY naive. A big part of her storyline is letting a Hitler expy escape, although he ends up dead anyway. When she later learns about World War 2 and what she almost allowed to happen, it nearly breaks her.
The cast of Homestuck are all thirteen years old or about that age. Sometimes they act in extremely adult ways, but other times they remind you that they are children. John watches R-rated action movies and has a huge crush on Liv Tyler (and arguably Nicolas Cage) but when Karkat implies Vriska likes him, he freaks out and says he's never felt that way about anyone before.
The titular character of Legend Of Maxx occasionally qualifies. He's generally stupid and has a tendency to Leeroy Jenkins into everything, but there are times where he proves to be somewhat competent, and he makes very elaborate buildings.
The Nostalgia Critic's intelligence and competence will vary depending on who he's with and if it's funny for him to be stupid or not.
In The Simpsons, due to being a 10-year-old boy, Bart fits the trope when it comes to things like sex, but that doesn't stop him from constantly making crude jokes and innuendo. In one episode, Homer sat him down and they had The Talk, yet in a later episode, this exchange occurred:
Bart: So you're not pregnant? But we held hands.
Darcy: Wow, you really are 10.
Bart: What a beautiful day. The sun is shining, the birds are singing, the bees are trying to have sex with them, or so is my understanding.
While this trope often applies to Bart, it should be noted that the episode where Homer gives him "the talk" takes place two years in the future. It wouldn't affect his knowledge of sex in the current day.
Likewise, Lisa switches between world-weary cynicism and childlike naivete as the plot demands. In one instance, within seconds - she hurls a bucket of red paint at Krusty (who's wearing a coat made from some kind of animal skins), screams "Fur is murder!" then follows it with "When's the Krusty movie coming out?"
And of course, there's Homer, whose borderline retardation is seasoned with odd Author Filibuster moments of uncharacteristic eloquence.
Dexter from Dexter's Laboratory had his moments of stupidity. One example is when he thought he was going to die...because he had gas. We're talking about a boy genius that has done everything from go inside his sister's body to traveling to other planets using his inventions.
To "simulate" the gas buildup in his lab, he used a balloon to represent himself. In other episodes/shorts, he didn't know what the "strange protrusions" all over his body were (he had the chicken pox) and tried to do a search through the entire animal kingdom for the "species" known as... the girl.
Stewie of Family Guy has always varied between someone that apparently had an adult's knowledge of the world, frequently showing knowledge of popular culture and sex (he once knew how his parent were going to make another child and understood exactly how the biological function was), and being fairly erudite but possessing little knowledge that a baby wouldn't (he didn't understand his birth even after his first birthday).
Also Brian is a really smart dog that can talk and nearly graduated college, but when he was tied to a pole with a rope he couldn't figure out how the rope had gotten shorter after running around it over and over again.
In "Halloween on Spooner Street," Brian discovers Stewie shooting at "monsters" with an assault rifle, only to ask him how he doesn't know what trick-or-treating is. Stewie asks Brian how he doesn't know that his reflection in the patio window isn't another dog.
Fry of Futurama also falls into this trope, with both his overall intelligence (varies from slightly below average to Too Dumb to Live) and his understanding of new inventions in the future.
Zoidberg's medical knowledge qualifies. In some episodes, Zoidberg is unable to tell the difference between humans and robots or know if a human is alive or dead. However, in one episode, he is able to save a mortally wounded Fry by attaching his head to Amy's body, and then repairs the body.
Zoidberg(at the wheel): "I'll just turn the wheel to maximum fastness!"
Starfire: "Your Earth ways are strange to me. Please, what is this 'so-fa' of which you speak?"
The boys on South Park, especially Cartman, are known to exhibit this.
"What's to understand? You get a boner, slap her titties around, stick it inside her and pee."
In "Marjorine" the boys thought the girls' paper fortune teller could actually predict the future, yet they were able to build and operate hi-tech machinery to study it with.
In "The Return of the Fellowship of the Rings to the Two Towers," Butters sees a good portion of what is apparently "the hottest porno ever made," yet many years later in "Sexual Healing" he sees a picture that includes a naked woman and becomes obsessed with it, even saying he's "never seen that part of a lady before!"
Specifically, he was obsessed by her pubic hair, something he probably hadn't seen in the porno.
"Scott Tenorman Must Die" - Cartman knows what pubes are, but not that you're supposed to grow them yourself (instead of purchasing them) and gets tricked into giving Scott Tenorman money and travelling to the "Pube fair" but is able to get revenge by tricking Scott Tenorman into getting his parents killed and then eating them.
The boys don't know that you don't have to wear a condom if you don't have sex.
In one episode, Jimmy remarks that he and his girlfriend took ecstasy and stayed up all night having sex. Later on, in "Erection Day", he gets his first erection and doesn't have a clue what he's supposed to do with it.
Almost the entire second season of Drawn Together was dedicated to doing this with half-retarded Ensemble Dark Horse Captain Hero. The sheltered Princess Clara is an equally big offender.
XR from Buzz Lightyear of Star Command is savvy enough to sneak into a large casino and cheat it out of a substantial amount of money, yet he willingly hands over all his petty cash to a street hustler.
In My Life as a Teenage Robot, Jenny's childish naivete and Brad's constant hovering between a savvy, nonchalant, typical teenager and a brash, bullheaded, over-eager, hero-wannabe.
SpongeBob SquarePants and his best friend Patrick also fall into this category. Spongebob varies from being pretty smart to being completely clueless that Squidward doesn't appreciate his company (although his problem may be more naivety than lack of intelligence). Patrick varies from being a Genius Ditz to being Too Dumb to Live. And while Spongebob is usually smarter than Patrick, there are a few episodes where Patrick is actually the smart one.
Hank Hill has this about certain issues. In one episode he mentions a bank teller who is "in between genders" indicating some sort of awareness of the issue, in another when Peggy tells him that her new friend Carolyn is really a man (drag queen though as opposed to transgender) Hank is genuinely baffled by the very concept, reacting with "Now wait a minute, that doesn't make any sense!" and continues to think of Carolyn as a woman for the rest of the episode, even going so far as to believe that, without her female clothes, she must be "Carolyn's boyfriend."
Both Fanboy and Chum Chum cater to this in various episodes. In one instance they're ridiculously annoying someone with their ADD-riddled antics, and later they're acting perfectly normal, usually when having to deal with a problem caused by someone that's not them.
Otto and Twister from Rocket Power, with Twister being more the naive one. One particular episode lampshades this in effect; the gang convince Eddie (Prince of The Netherworld) to skate in his empty pool as long as they keep it to themselves and leave before his parents get home. Twister, for his part, keeps quiet, even when his brother Lars comes to make mischief. Otto on the other hand, kept bragging about it to strangers like it was a party, even to Lars. Reggie's What the Hell, Hero? one liner sums it up perfectly.
Reggie: Did you and Twister switch brains?!
Animaniacs has Skippy Squirrel, who switches between an innocent Foil to his Aunt Slappy and a companion to her shenanigans whenever the episode calls for one or the other.
My Little Pony: Friendship Is Magic: Not naivete, per se, but Spike's emotional maturity fluctuates quite a bit depending on the episode. For example, compare how he acts in "Lesson Zero" (where he plays the Only Sane Man to Twilight's neuroses) to how he acts in "It's About Time" (where he takes advantage of a similar situation to mess with Twilight and gorge himself on ice cream). This goes hoof-in-claw with his exaggerated Vague Age, as despite occasionally being called a baby dragon, he also works full time as Twilight's research assistant and valet.
Other characters display related tropes, but none (Except maybe the Cutie Mark Crusaders) quite qualify.