In fiction, being overweight doesn't necessarily mean you're probably a bad person...
Maybe you're just stupid. In any group of characters the fattest one will probably be the dimmest.
Granted poor diet and a sedentary lifestyle is not necessarily a clever lifestyle choice (leaving aside more complicated real life reasons for weight gain that are unlikely to be portrayed in fiction), but many tubby characters seem to think of literally nothing else other than eating. Even if they are good, the fat character is generally a comic ditz.
This trope is particularly popular in animation.
See also Fat Bastard, Fat Slob, and Shorter Means Smarter. Often an element of a Straw Loser. Contrast Lean and Mean.
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Anime and Manga
Gluttony from Fullmetal Alchemist is basically a pet ball of fat who consumes everything in his path. It takes Alphonse Elric about 30 seconds to convince Gluttony to lead his enemies right past all the defenses in the headquarters and right up to the Big Bad. Alphonse and Edward's presence wasn't especially minded, since they're two of Father's candidates for "sacrifice", and needed to be kept alive; everyone else, on the other hand...
Averted with Breda, who graduated top of his class.
On the other hand, he acts a lot less serious/intelligent than the rest of Roy's crew. Then again, he is the Deadpan Snarker...
Chaosic Rune: About users who can summon powerful "dragons" to do their bidding. One of the most powerful is Death Rex, whose limbs and body got separated from its main "head". Death Rex's "body" (who has taken possession of a human body) , is a lumbering brute who is not too bright and is obsessed with eating. Much to Death Rex's chagrin, his "body" has gained a bit of weight - . "For disgracing my plump body! I'll beat you to death!"
Sumiyoshi of Excel♥Saga is a subversion. He may not speak at all, but he's still probably one of the smartest and most normal people in the entire cast.
Invoked and then averted in Basilisk. On the surface, Jousuke is this trope and then some more, but it's actually Obfuscating Stupidity: he pretends to be dumb, but when push comes to shove he turns out to be quite the Lightning Bruiser.
Arnold of Soulless Shell, overlapping with Fat Bastard. He falls for Leif's ruse to get him to admit to being guilty of rape when he sentences him to death, then after breaking out of prison, plans on killing Leif in revenge before going off to rape a girl. Since Leif is a GodModeSue with a particular hatred of rapists, Arnold dies a fairly nasty death.
Arthur in A Very Special Arthur, who becomes retarded during the course of the story and is also described as quite overweight.
Subverted with John Candy (also now deceased), although in some he played a Bumbling Dad-type character, the character he played all had their heart in the right places. He was quite an intelligent guy in real life.
Piedone sometimes subverts and sometimes plays this straight. In a movie where his role is that of a higher up policeman (so we're expecting him to have some brains), he's tangled in a web of corruption and doesn't know what to think about the situation anymore. In a So Bad, It's Good scene basically summarising Piedone's entire film career, he says "What Piedone doesn't like, Piedone punches!" and proceeds to punch his superior into unconsciousness. The guy turns out to be legit and even help Piedone in a tight spot.
This attitude shows up in Robin Hobb's Soldier Son trilogy, including the difference in values. Among the Gernians (the "civilized" people), obesity is regarded as a sign of lack of self-control. Among the Specks (the "nature people"), it's thin people who are seen as stupid. The attitude is, "What kind of idiot can't even provide for himself?"
This trope is largely averted in the Discworld books as the wizards are mostly very fat while being very clever (if not wise). On the other hand most of the more dynamic, forward looking wizards tend not to be fat such as Ponder Stibbons and (in a very different way) Ymper Trymon.
Sergeant Colon is a straighter example, though he's portrayed more as an average man on the street than a moron.
Vetinari: The entire success of this plan depends on you seeming to be a fat idiot!
Sgt. Colon: I'm not too good at acting, sir.
Played very straight with Crispin Horsefry in Going Postal, complete with a footnote expressing the author's irritation that people seem to believe in this trope in real life.
It sometimes seems like Nanny Ogg has intentionally invoked this trope in her old age, allowing herself to get "larger than life" as part of her image as a witch (that she is the one usually present at a birth rather than a death and that she is kind-hearted and jolly but not necessarily bright).
King Rhodar of Drasnia in The Belgariad is also a subversion. His obesity is the result of loving books and scholarship more than hunting and warfare. His personal library is the third largest library in the world, and he is shown to be an excellent administrator. Even the more martially inclined rulers defer to his expertise in strategy, due to his excellent theoretical background.
Kevin from The Office, especially after Flanderization sets in. Before it he was revealed to be a World Series of Poker bracelet winner, which implies a level of competence with numbers and probabilities that the Kevin in later seasons couldn't hope to accomplish. To give an example of just how stupid he's become, at one point he was unable to distinguish between a coin and a button.
Hurley from LOST. The trope was played with as various handwaves and justifications propped up to describe his supernatural clumsiness and misfortune.
I think Hurley might be an inversion; although his ideas and opinions were often dismissed in the early seasons, he invariably turned out to have more common sense than most of the other characters. And he did end up taking charge of the Island and it's implied that he did a fine job with it.
Hurley was originally supposed to be an idiot. His first appearance in the original script describes him as a "fat, obnoxious Red Shirt''. But the creators ended up liking him, and kept him around. The guy who dies in Season One while holding a stick of dynamite and warning them how sensitive it was is the character Hurley was originally going to be.
Possibly averted with Gordon on Freaks and Geeks. While we never really see him demonstrate intelligence, he is far and away the most thoughtful and insightful of the geeks.
Chumlee from Pawn Stars is portrayed this way, however, occasionally he proves to be more clever than his reputation might suggest, and he's not the only overweight character on the show.
Sergeant Garcia got this treatment in Disney's Zorro series.
Larry Kubiac in Parker Lewis Can't Lose isn't really fat but just really, really huge. In the first few seasons, he is depicted as a typical high-school bully who only thinks of eating. Later, he becomes more of a gentle giant and it's revealed that his simple-mindedness is just a facade and that he is in fact fairly intelligent.
The Colonel Blimp cartoons by David Law (though not the 1943 movie, which is In Name Only).
Subverted in Final Fantasy VIII. Ward is roughly as intelligent as most of the other characters (except maybe Quistis), and provides most of the words of wisdom for his circle of friends (except for when, well...you know.)
Ace Dick from Problem Sleuth. He's fat, and his imagination is handily the worst of any character in the comic. To illustrate, while Problem Sleuth, Pickle Inspector and Mobster Kingpin can all imagine fairly convincing female representations of themselves, Ace Dick can only imagine himself with a crappy wig on.
Eric Cartman from South Park. Though his hatred and spite occasionally give him moments of brilliance.
To elaborate, Eric Cartman can be seen as this, but in truth, he's not stupid at all, at least when it comes to how society works and not academics. He is quite knowledgeable in lots of trivial subjects, in all honesty, but he's also arrogant, spoiled, rude, racist, sexist, and unstable. On top of those, he's also very, very, very shortsighted and has an incredibly low attention span, which are the prime reasons he can be seen as an idiot. Just keep in mind that whenever he goes into Tranquil Fury, that's when he loses those two traits.
Jack Fenton in Danny Phantom. He's often portrayed as dim-witted, an incompetent ghost hunter, and slow to catch hints. But then, he also has several Moments Of Awesome that occasionally remind you that he is a scientist.
The unnamed Giant from the Looney Tunes short "Jack Wabbit and the Beanstalk" may very well be the fattest and stupidest enemy Bugs Bunny ever faced off against. Here are a few examples of his idiocy: he's thinks he's smarter than Bugs because he's a moron, he spells smart like c-a-t, it took him about 5 hours to realize that Bugs wasn't coming back in the jar he trapped him in after escaping, and it took him about 20 seconds to realize his head was on fire.
There is also Rocky's henchman Mugsy.
Also Little John in "Rabbit Hood". And Junyer Bear.
There may be some Truth in Television since a five-year study in France has found a possible link between weight and brain function. People with a higher Body Mass Index (BMI) scored lower on a cognitive test than average people. This has been nicknamed the "Homer Simpson effect". See for yourself.