Cartman perfectly modeled this trope once when he was caught in class mimicking certain Nazi gestures and phrases, and was made to watch an anti-Nazi film, which only fueled his fascination with the subject. Considering that the film only showed Nazis marching and never mentioned any bad things that Hitler did, only that he "was a very naughty person", and considering that Cartman's anti-semitic and a sociopath... yeah.
In addition to this, we had the "Major Boobage" episode with the sub-plot of Cartman rescuing the cats from being put into the pound (eventually taking in around 100) - even going so far as to suggest that Mr. Kitty "Write a journal", as well as replying "They're innocent victims in this! They have to hide or they'll be put to death. Something you just can't understand." when asked why the cats are in his attic. By the end, we have this exchange between Cartman and Kyle
Cartman: But ya know, we've all learned something, you guys. We can never persecute living beings and force them into hiding. It's wrong. Kyle: And you don't see any parallel between that and anything else in history? Cartman: Hmmm, nope. I have no idea what you're talking about Kyle.
In the episode where Wendy fights Cartman, Cartman is terrified that Wendy might actually be able to beat him, which would be terribly uncool. He manages to get her into enough trouble so that she'll be severely punished if she goes near him, effectively defusing the situation... at which point he starts taunting her again, which makes her even more determined to fight him. Repeat about three times. When they finally fight she beats the crap out of him (even declaring "I'm finished!"). When Cartman gets back onto his feet he tells the boys that since he's been beaten by a girl he'll be uncool. The boys tell him they never thought he was cool to begin with. Cartman decides that since this is obviously impossible, the boys are lying to spare his feelings because they think he's so cool.
An example of Comically Missing the Point that's Heartwarming: Chef finds the flag of South Park (depicting four white guys lynching a black guy) offensive for some reason. In an epic buck-passing motion, the Mayor decides to delay the vote until the kids have a debate on it. When Stan and Kyle say they're in favour of keeping the flag as is, Chef is naturally dismayed. The debate arrives, and the main point of Stan and Kyle's argument: Killing is natural, so changing a flag that has a guy being killed is pointless. Chef explains that this isn't why he finds the flag offensive, and explains the real reason, to which Stan responds "But...why does the colour of his skin matter?"
"Broadway Bro Down" has Randy attending a Broadway play with his wife and he learns from someone that musicals have subtext hinting about sex every 10 seconds. The subtext apparently causes all the mens' wives to give them blowjobs. Randy listens closely and hears blowjobs during the scenes which is mixed in with the normal lines. Randy takes the entire thing out of context by creating a play where the implications of blowjobs are plainly said instead of being subtle.
When KFC is outlawed in Colorado at the same time that medicinal marijuana is legalized, a bunch of men deliberately dose themselves with radiation to get testicular cancer so they'll be able to legally obtain medicinal marijuana. Meanwhile, a cartel responsible for smuggling and selling black-market KFC (which is depicted as highly addictive) has brought a huge amount of crime with it. However, the crime influx is mistakenly blamed on marijuana being made available, and the town doctor decides that since cancer rates shot up when South Park's KFC was replaced with a medicinal marijuana dispensary, the KFC must have been protecting everyone from cancer somehow. Thus, against all logic, the town successfully solves its problems by once again criminalizing marijuana and legalizing KFC (renamed Medicinal Fried Chicken).
Cartman is revealed to be Scott Tenorman's half-brother in 201. Cartman gets upset not because he killed his own father (and turned him into chili), but because this means that he's half-ginger.
In the episode where Cartman wants to become a Nascar driver, he thinks he can't because he isn't "poor and stupid enough," and after he crashes a car and kills 11 people, he thinks it's because he's not poor enough.
In this episode, it had multiple people insulting him (Kyle telling him he's a broke, ignorant retard, the doctor saying that it was the most idiotic thing he had ever seen anyone do) and Cartman replying with "Thanks for trying to make me feel better, it's not going to work." Even in the end, when he loses, he just simply realizes that he will never be poor or stupid enough.
Kenny tries to sneak a sniper rifle into the track to shoot Cartman for making Nascar drivers look like poor dumbasses, and the security guy simply says "You're the type of person that makes us Nascar fans look like idiots."
In "Cartman's Incredible Gift", the police believe Cartman is psychic and can help find a serial killer. After arresting the person Cartman has fingered, another person is murdered in an identical fashion. The police conclude that a copycat killer is on the loose and need Cartman's help. This continues...
Sgt. Lou: They just found another body! That means a fifth copycat killer is on the loose! Where's my psychic?!
In "Fishsticks" Jimmy comes up with a joke while Cartman sits on the couch. Cartman then believes he came up with the joke and when he sees Carlos Mencia claim to be the originator, he angrily tells Jimmy that some fat turd is taking credit for something he didn't do. When he asks Kyle for advice because he thinks Jimmy is taking credit away from him, Kyle explains that people like Cartman have such big egos that it makes them think they are awesome when they are not. Cartman simply thinks that Kyle was trying to warn him about Jimmy.
Kanye West has such an inflated ego that he thinks the fishsticks joke is specifically about him. Kanye misses the point of Cartman's epiphany at the end of the episode and still thinks the fishsticks thing is about him, now believing he was really a gay fish in denial, and "returns" to the sea to have sex with male fish. A deleted scene on the DVD shows he drowned.
Most of the world seems to have this problem in "Dead Celebrities" when it comes to food either from the Chipotle restaurant or made with chipotle peppers (they're just vague enough to get away with the accusations). Apparently, people (in the South Park universe, anyway) tend to suffer anal bleeding from eating such foods. However, the only product offered to deal with this is a substance that cleans bloodstains out of underwear. Kyle is the only one who thinks that the problems of anal bleeding might go beyond blood-stained underwear.
In "Coon vs Coon and Friends", despite teaming up with Cthulhu and leaving a bloody wake across the country, Cartman still continues to see himself as The Hero and Kyle and the others are villains for trying to stop him from "saving" the world. When the others point out that all he's doing is making the world better for himself, he states his belief that's what superheroes do?!
In "Death," the earlier airings feature the following exchange:
Liane: "Eric, dear? I just got a call from your friend Kyle's mother. She said that this show is naughty, and might make you a potty mouth."
Cartman: "That's a bunch of crap! Kyle's mom is a dirty Jew!"
Liane: "Ohhh, okay, hon."
Later airings censor the word "dirty," making the joke seem less like Cartman's mother is ignorant of her son's racism, and more like she's just plain stupid.
In "Death Camp of Tolerance", Principal Victoria is so obsessed with being tolerant that she ignores all the horrible things Mr. Garrison does to try and get fired, up to and including shoving the class' gerbil up Mr. Slave's ass and doesn't do anything until Mr. Garrison calls her out on stage.
And even then, she responds by sending Garrison and Mr. Slave to tolerance camp, as they obviously don't tolerate themselves!