In Human Curiosity, America pretends to be tactless and unaware that he's pissing off Arthur Kirkland, in order to pull off Arthur's sunglasses and see if he really is England. It's noted that one of the good things about America tending to be The Ditz is that no one notices the rare instances when he's only pretending to be stupid for his advantage.
Kyon, in Kyon: Big Damn Hero. He is shown to be highly fluent in English, easily beating the brigade in Scrabble, very skilled in chess*
Asakura's best against him was a single stalemate
, and invokes Gratuitous English. He also learns German from Tsuruya extremely quickly. Aside from language skills, Kyon also managed to convince a renegade time traveler to pull off a Stable Time Loop for him, or risk being killed, because Kyon can go back in time to take his place, thus rendering his Contractual Immortality moot.
Cruise Control in Turnabout Storm. This particular case is more Played for Drama than usual, as they had seen themself more or less forced into it thanks in part to Ace Swift blackmailing Cruise into losing to him. Cruise thought the whole act would be more believable if he thought and acted like a general loser as well.
Light: One does not just randomly select a question or 2 and try to get them wrong. No. Nor do you do it to every 4th question or some other such ridiculous, predictable pattern like that. The key lies in, like all things, strategy. You need to choose ahead of time a subject or 2 you're awful at, a few you're neutral, and 1 you're good at. Then whenever you get homework, consistently get the same basic principals wrong, and get it wrong on the test as well. Also, you have to make a show of not paying attention in class, or else suspicions will be raised, and you'll be given a special "learning disability," and assigned a tutor pronto.
A few fanfic authors like to play with the idea that Richard Moore is aware of what Conan is doing, or faking being asleep. Has a basis in the simple fact that being injected countless times with the same sleeping drug will eventually result in the body resisting it.
Crabbe and Goyle are quite commonly described as doing this. Sometimes they are just shy or quiet, and that combined with their appearance, their love for food and their status as Draco Malfoy's cronies make people think they are stupid. Other times they are actively pretending to be stupid so that people won't expect anything of them. In both cases they either actually like Draco and he likes them, or they protect Draco physically (as is implied in canon) and he protects them socially. In one particular fic, they weren't stupid, but since they suffered from undiagnosed dyslexia, everyone thought they were.
Kakashi in From His Excerpts by Phoenyxx. Kakashi speaks fluent English, but hides it from the professors, resulting in little gems like these:
Oh. Professor Flitwick chuckled lightly, "I believe the correct term is squib."
"Yes, squid. It is what I say."
In Legacy of the Rasengan Naruto is a creator of simple-to-use yet effective jutsus (simple as in easy to activate and tweak), thinks before he acts, is very observant but acts like his canon self to hide it. The only ones who know (at first) he is playing dumb are Kakashi (due to Naruto outwitting him during the bell test and carefully observing Naruto "Take in information, process it instantly, then act in a manner expected of his buffoon self." ), Hinata (he doesn't put up the charade when they're alone together) and Shikamaru (connected the dots like Kakashi). As is common with the trope they call it dumb luck if he pwns with wits/acts unusually perceptive and jaw-drop when he shows jutsu. Everyone eventually figures it out during the Chunnin Exams after he displays a jutsu that works like Jyuuken andCLEAVES THROUGH NEJI'S ROTATION!
Kaede Mizuno in A Growing Affection operates using this trope, she wants her enemies to think she is just a pretty face who was promoted due to her family connections. She is in fact quite intelligent, and goes toe-to-toe with some of the strongest characters in the story.
In With Strings Attached, the four all pretend to be naïve, useless fools when they first start journeying with the Hunter, because they take a massive dislike to him the moment he enters the scene. Their intent is to gradually reveal their power as they travel so they can humiliate him. This turns out not to be the best idea they ever had.
During the late '90s, Usagi of Sailor Moon practically ruled this trope. It was all the rage for fanfic writers to make Usagi into a God Mode Sue who was only pretending to be an idiot, often by leaving the Inners and dropping the act. That she still can't write proper hiragana as queen and routinely swipes her kid's candy in the canon is conveniently ignored.
She is The Ditz in the anime, but shows that she is actually pretty smart in the manga, especially as it goes on and the burdens of leadership take their toll. Venus can be considered another case as well, since it is hinted that she speaks fluent English, a language that is just as hard for the Japanese to learn as their own language is for native English speakers. (They're in completely different language families, so have no common baselines).