I've seen, been involved in, and sometimes caused, many a facepalm moment via what the played characters do to each other and to the world around them...and this is all (usually) good fun.
Where it gets jiggy is when the players take the arguments out of character - I've seen this too, and it ain't fun at all.
Worst one I've ever had to DM: I was running two side-by-side parties in the same world, one on Sundays, the other on Mondays. One night a Sunday player (not universally liked by the Monday crew but someone all had known for ages) stopped by the Monday game to visit, and was half-jokingly given an invitation to "roll up a character, so we can kill it". Well, he rolled up a character - a Paladin that nobody would have any good reason to kill - and brought it in to see how long it could last. In hindsight I shouldn't have allowed this, but hey...I was curious to see what would happen next, and everyone was still laughing. And sure enough the party's resident Assassin (also its longest-serving character at that point) took down the Pally at the first opportunity. The laughter stopped.
The resulting out-of-game argument went on for several weeks.
Worst one I've hit as a player: about 25 years ago one of my characters killed another PC with a fireball - he in-character claims I hit him intentionally, I in-character claim he ran into my pre-defined blast zone*. Not only did he die but several of his magic items went boom. We get back to town, and he gets revived (for free, someone else paid for it). He then takes my character to court to reclaim all the expenses and losses caused by my fireball...including the cost of his raise. He bribes the jurors and I lose big-time; I had to retire my character so she could pay off her debt in time as she didn't have anywhere near the money required.
This was also good for an out-of-character argument that went on for ages.
* - in fact both were sort-of true; he knew where I was aiming and ran in anyway, I knew he was there and took the attitude of "if you're going to be that stupid, so be it" and held my aim.
As fate would have it these two characters met again many years later in a reunion game - let's just leave the results at I survived and he did not.
Lan-"and the fallout from that reunion game still hasn't all hit the ground"-efan