Character mistake: When Chief Wiggum shows up to arrest Nelson during the game at the end of the episode, Bart goes with Wiggum instead while pretending to be Nelson. Even a cop as incompetent as Chief Wiggum would not have mistaken Bart for Nelson since he'd had numerous dealings with both before this episode; Wiggum may not be particularly bright, but he knows who's who in Springfield.
Suggested correction: Not really a mistake for this show in particular. Characters like Homer, Bart, Chief Wiggum, Grampa and others are deliberately written to be on a sliding scale of stupidity, pretty much exclusively for the benefit of whatever joke the show is telling.