Continuity mistake: At the beginning of their first pool match, Fats' collar goes from buttoned to loose, to back buttoned in only a moment on-screen.
Factual error: In the first match between Fast Eddie and Minnesota Fats towards the beginning of the film, Eddie, right after declaring he would start to win after losing thus far, shoots the one ball into the side pocket by shooting the cue into a striped ball. Sharp eyes will notice that he actually hits THROUGH the cue and strikes the stripe into the one with his stick. While a nice trick, this is an incredibly illegal shot and would never have been allowed.
Other mistake: When Fats makes the 7 ball after Eddie's safe play, you see the cue ball flying into the pocket before the camera changes shots.
Continuity mistake: When Newman first encounters Laurie at the diner in the bus station, in a wide shot we see all the empty tables behind her. He sits down and their conversation is shot in medium close-ups. In all of Laurie's shots, a man is now visible three or four tables behind her, flipping through a newspaper. When the conversation ends, we go to a wide shot, and, the guy has vanished. About ten minutes later, next day in the time line (I think), another scene in the same diner. Newman is already seated as Laurie walks in. She walks past empty tables to Newman. A few shots later, suddenly, in the background, the guy is back at his table with his newspaper! As they walk out the guy is gone again. It's the same guy at the same table appearing/disappearing in two scenes.
Continuity mistake: In the scene depicting the first game between Eddie and Fats, having just broken off in the first frame, Eddie chalks his cue with his left hand while his right hand is placed towards the butt of the cue. In the next camera shot however (a split-second afterwards), Eddie is chalking the cue with his right hand with his left hand positioned on the upper half of the cue. As a result, the background scene contains a continuity error also - the man with the striped tie is seen putting a cigarette to his mouth in shot 1 but his cigarette is down by his knee in shot 2.