The story of Hamlet is based on the legend of the Danish Prince Amled, whose father was murdered by his Evil Uncle Fenge. Amled took to sitting near the fire and carving wooden hooks all day, telling everyone that he would use them to avenge his father. Fenge thought he was crazy - until the night where Amled used the hooks to pin down Fenge and his men under their sleeping blankets and burn down the palace over their heads before they could get free.
A female version of the legend has a man kill his tribe's chief and take his place, shortly before lusting after the former chief's daughter. Said daughter knew full well who murdered her father, but pretended to not know anything. Then the murderer asked her to meet him in his tent one night, and she brought a knife with her... And cut some bread?
According to his legend, Saint Simeon the Holy Fool. He was a simple ascetic monk whom God himself asked to act like a madman so he could save souls, and lo did he make honor to that.
One old story tells about a guy who owned someone else a big amount of money and would be thrown into debtors' prison if he couldn't pay. However, he found an Amoral Attorney who promised him to bail him out for four gold pieces, to be paid after a successful acquitting. So the lawyer advises his client: "When in court, say nothing but 'bleh!', whatever happens!" The process starts, and the client indeed answers nothing but "Bleh!" no matter what he's asked. Finally the judge has enough:
Judge: "Why does your client say nothing but 'bleh!'?"
Amoral Attorney: "I'm sorry your honor, he's an idiot, when I was talking to him, he also said nothing but that!"
So the judge comes to the conclusion that the defendant can't be condemned and lets him go. Now the Amoral Attorney demands his money. But the client, again, just says "Bleh!"
Amoral Attorney: "Are you joking? You promised me four gold pieces! I want them now!"
Client (tips on the table): "Bleh, bleh, bleh, bleh!"
The 16th-century Teutonic legend of the Schildbürgers says they were great sages who went far and wide to advise princes — until their wives got sick of them going far and wide and they needed to get the princes to stop seeking their advice.
According to legend, the Roman Lucius Junius feigned stupidity (earning the name "Brutus", Latin for "dullard") to avoid being killed by the evil king Tarquin. When the time came, Brutus dropped the mask and led the overthrow of the monarchy, establishing the Roman Republic. (Centuries later, his descendant Marcus Junius Brutus followed his ancestor's example by participating in the assassination of Julius Caesar.)
Socrates was so good at this, they had him killed.
One joke recounts the tale of a kindly shopkeeper and a little kid named Billy. On many an occasion, the shopkeeper would witness older boys teasing Billy by offering him a choice between a nickel and a dime, then laughing at him choosing the nickel, supposedly because the nickel was larger and Billy was too slow to realise that the dime was worth more. Eventually, the shopkeeper took pity on Billy, and took him aside for a quiet word on the matter... only for Billy to reveal that he was playing this trope all along: he knows very well how much the two coins are worth, but if he ever picks the dime, the kids will stop giving him free nickels.
Maria: Last week Bischoff abused his power in a way that was both malicious and capricious, and it’s this rash of discourse that ultimately led to a locker room of disdain and mutiny, and it should be grounds for his immediate dismissal.
She went back to the ditzy gimmick for quite a while afterwards. In a subsequent appearance, Bischoff flat-out accused her of using Obfuscating Stupidity based on what happened at his trial, saying she pretended to be "stupid and sweet" to curry favor from the crowd.
LayCool could count. A pair of bubbly ditzy Valley Girls who have managed to dominate the division for over a year, capturing four sets of titles for themselves.
During a brief feud with Maryse, Gail Kim appeared oblivious when Maryse started talking to her in French, appearing to be trying to be friendly. Anyone who understood French knew Maryse was secretly trash talking and Gail looked to be falling for it. Then Gail dropped the bombshell that she spoke fluent French and hadn't been fooled by the game.
The late English magician/comedian Tommy Cooper was an undeniable master of using Obfuscating Stupidity in his magic acts. He intentionally botched his own tricks and acted incompetent 90% of the time until he'd pretend to foul up yet another, only to pull off the trick perfectly.
Ray Romano suggests that guys screw up shopping as badly as possible so they're never asked to do it again. "They were out of lettuce, so I got a hammer."
Similarly, in "Chocolate Cake for Breakfast", Bill Cosby so horrified his wife by going along with the titular request that she told him to go back to bed.
Bill: Which is where I wanted to go in the first place. So you see? We are dumb, but we are not so dumb. It takes great thinking and work to keep from working.
Larry The Cable Guy. In-character, he's a not-too-smart Southerner. Out of character, he's Daniel Whitney, a reasonably intelligent satirist. This trope is best exemplified in his book Git-R-Done, where he makes deliberate typos as a form of Stylistic Suck, yet launches into Shown Their Work territory whenever he rants about something that bugs him. (To name just one example, he explains Darwin's Theory correctly.)
In the Legend of the Five Rings franchise, this is the basis of the Scorpion Clan's modus operandi, as revealed to their founder Bayushi by the great sage Shinsei. Shinsei told Baysuhi a parable that Bayushi believed he already knew, about the scorpion and the frog, which normally ends with the scorpion stabbing the frog while they're crossing the river and both of them drowning because it's the scorpion's nature (this parable also appeared in the Star Trek: Voyager episode 'Scorpion'). However, at the end of Shinsei's version the scorpion's answer is different: "I can swim."
The Space Wolves in Warhammer 40,000 are not as stupid as they act, especially their Chapter Master Logan Grimnar. There is a reason he ends up running most of the wars he gets involved in, even if its from the sidelines.
The Horus Heresy novels place the Legion in a new light. Leman Russ frequently puts on the persona of being the rough barbarian king. This isn't just a convenient affectation for political purposes; it's also a coping mechanism for the things that he has to do. In a similar way, the Space Wolves socially act like ancient norsemen but are plan their attacks with complete and total precision.
Likewise Ciaphas CainHERO OF THE IMPERIUM isn't so much "stupid" in his obfuscations, but he makes it clear in his monologue that he needs to pretend to be much more of a gung-ho Emperor-lover than he actually is, if he hopes to stay alive and maintain his heroic reputation.
Although most of the time Homestar Runner seems just plain dumb, in the Strong Bad Email "stupid stuff", he tricks Strong Bad into using reverse psychology in order to win a bet and make Strong Bad lose his.
In The Salvation War, Jesus first appears to be a shiftless stoner. Then the angels leave, and he reveals himself to be a skilled strategist as he improves on the battle plan the angels gave him.
In Sims Big Brother 6, Alison's strategy was to appear to be incredibly stupid, so people wouldn't think she was a threat and never nominate her. She unfortunately went a tad far with it, since managed to misspell "Apple" and people begun to suspect whether or not this was an act or not.
It was also surprising when she suddenly spouted a college-level description of an organelle after a montage of stupid questions. ("Controversy? What does that mean?" "Influenza? What's that?" "How do you work the elevator?")
The Nostalgia Chick during Kickassia, where she went from sane (or as sane as she can be) and opinionated to a Sarah Palin parody who agreed with N. Bison about everything. When Part Four came around, it was found out she was just pretending to be sweet and stupid while trying to kill him off and take charge for herself.
And then there's Kevin Baugh, who fakes being taken over by his alternate personality so he'll be allowed to keep hanging out in his conquered nation, then sets to planting doubts about the Nostalgia Critic's leadership skills. Bonus points that everyone knows he's doing this but roll with it anyway.
Dragon Ball Z Abridged turned Maron of all people into this. She was really an undercover agent investigating Krillin for insurance fraud. Still a Gold Digger, though.
Sometimes Akinator will ask a logical series of questions leading to your character, then suddenly start asking random unrelated questions, apparently having been thrown off the scent. In the end though, he gets the character right. He figured out what your character was early on, and was just using the remaining questions (and you) to learn more about the character in question.