Continuity mistake: At the end of the episode, watch the colour of the time-machine inventor's outfit. You can see it's light blue when he pulls the lever, but when he gets the Flintstones and Rubbles back to the fair and asks them what happened, his outfit is a darker shade of blue. It then changes back to its lighter shade as before.
Continuity mistake: In the scene where Fred and Barney are in the doctor's surgery, the doctor produces a lollipop for Fred. When he produces the lollipop, you can see it's well away from Fred, but as the shot changes the lollipop is much closer to him than it was in the previous shot.
Continuity mistake: In the scene where Mr. Slate asks his employees to gather round for a group photo, Fred decides to liven up the situation by giving Sandy a 'hot foot'. When Mr. Slate takes the photo, Fred is lighting the match on Sandy's foot, but the picture that comes out of the camera shows him not lighting the match at all.