Inspector Jacques Clouseau from the original Pink Panther movies is only able to solve a case through sheer luck. He'd often be moving away from whatever the clues are pointing to, it's just that in the end, the puzzle all falls together in his favor. Steve Martin's Clouseau is more of a Genius Ditz than The Fool, though.
In A Shot in the Dark, he irrationally refused to consider that Maria Gambrelli was behind all the murders because he was madly in love with her, despite the fact that all the evidence points to her doing it. At the end of the movie, the mansion's other occupants get in a heated argument and all of them accidentally confess to a murder each. Realizing that they're about to be arrested, they tried to escape in Clouseau's car, which incidentally was rigged with a bomb installed by his insane boss Inspector Dreyfus. The case is solved, Clouseau is alive, and the murderers all suffered Karmic Death.
There was even an early wild guess that he's secretly a Force-user, because he's suspiciously lucky:
Jar Jar Binks takes out droids as fast as Qui-Gon and Obi-Wan do, and he's not even trying.
In an episode of Star Wars: The Clone Wars, a Clone Trooper captain has apparently been Jar Jar's bodyguard long enough to accurately predict when the Gungan's antics will defeat the enemy. However, the captain wrongly believes that Jar Jar isn't as stupid as he seems.
There was a rumor that Jar Jar's character development was supposed to be highlighted into a transformation had the fan base not hated him. He was actually supposed to become a tragic character - becoming a Dark Jedi out of his grief from Padme's death and Anakin's movement to the dark side.
Scrat from Ice Age, although he's half fool and half desperate for nuts.
The Man Who Knew Too Little. Wallace Ritchie believes himself to be in some form of improvisational theater, but is actually involved in a cold-war era plot to prevent peace between the UK and Russia. While in one or two incidents he does show some genuine skill, most of his success is due to sheer dumb luck. The movie is even funnier if you try to watch it from the point of view of the real people, who must be assuming that Wallace is merely playing The Fool, while actually being a top-notch spy. So The Fool is playing the Spy playing The Fool.
WALLE: WALLE changes humanity's future, returns the human race to Earth and gets them out of their lethargy. He meant none of it, for the entire movie, his sole goal is to hold hands with a girl he met.
The Party has another, less known Peter Sellers example, in which he plays the sweet but incredibly accident-prone Indian actor Hrundi V. Bakshi.
The Bobo is an even less-known Sellers vehicle, where he plays an itinerant singer who tries to get his big break at a Barcelona theater (note: 'bobo' is Spanish for 'fool'). The theater owner agrees on the condition that, within three days, he can seduce a manipulative gold-digger. Trope averted as Sellers is shown to be adept at concocting and sustaining an elaborate ruse to string her along.
The Tall Blond Man with One Black Shoe is an innocuous concert violinist who, unknown to him, is identified as a spy as part of an intelligence agency rivalry. He walks through the film oblivious to the machinations of the agents keeping tabs on him, and remains unscathed as they do each other in.
Danny Kaye played this role in The Inspector General, as a hapless, illiterate, but good-hearted gypsy who is mistaken for an Inspector General (on assignment to root out corruption from Napoléon Bonaparte himself) by a small town's corrupt city council.
Ironically, although he skirts it at times, he doesn't play this role in The Court Jester, as he has some idea of the danger he's putting himself through, but accepted because he wanted to be a hero. And impress Maid Jean.
Mose Harper in The Searchers has been described as a kind of "holy fool" by some film critics. Throughout the movie he tends to be looked on as a crazy old coot by most other characters, yet he gets safely out of dangerous situations, even out of being captured by the tribe of the not exactly merciful Chief Scar ("made 'em think I was crazy"), and in the end supplies the information that brings Ethan Edwards' search to its conclusion. The only thing Mose himself wants in life is a rocking chair to sit in, and at the film has achieved that goal...
Sodnom, the village idiot in A Pearl in the Forest, set in 1937 Mongolia during Stalin's tyranny. He grins stupidly at everyone, laughs inappropriately, and generally comes off as being mentally retarded. The ending reveals that it's a case of Obfuscating Disability, a survival tactic to get him through the purges and oppression of Stalin's rule.
Fatal Instinct: Lawyer/cop Ned Ravine wanders blisfully through life as an ignorant buffoon, not noticing that several different people are all plotting his death.
Ned: I don't look as dumb as I am.
Harry Chelm is Beat the Devil a classic type. He goes about in blissful ignorance, completely missing his wife's affair with Billy until she tells him, blowing off all of Billy's well-meaning advice. Once the bad guys steal his attache case onboard the ship, he starts barking threats about how he's going to turn them all in when they make landfall, in complete obliviousness to the fact that the bad guys thus have every reason to kill him. Sure enough, they try to, and Harry is just barely saved at the last second by Billy. Gwen then labels him a paranoiac in order to save him from being murdered, but he is put in irons onboard the shipwhich leads an offended Harry to jump ship and attempt to swim to land. And he makes it, and takes possession of the land that the bad guys wanted, and strikes it rich. Even better, the name "Chelm" is almost surely a reference to a legendary Jewish town ruled by "wise fools" who come up with silly solutions that usually somehow work out and fix problems.