Continuity mistake: After Jim Carrey crashes the car on the bridge and, we see in the long shots that the lamp on the half-broken post above him is swinging from side to side. However each time we cut to a closeup of Carrey's face, it is lit very evenly - although the only source of light nearby is swinging wildly.
Factual error: The film is set in 1951 but Peter and Adele have their first dance to the tune "Stranger On The Shore" which was first released in 1961.
Factual error: In the very end, when Peter is returning to town, as the train rounds the bend you can see much newer style electrical transformers on the power poles. If you look close you can see a modern style street lamp also.
Factual error: The locomotive used to haul the train that takes Mr. Appleton back to Liberty after facing judgement is an Alco Century series - which weren't around in 1951. Nowadays, the locomotive type is actually semi-vintage, but it is obvious that this particular one has been patched up and modified over the years. It has intercoolers, ditch- lights above the pilot and a blank nose where the lower set of headlights used to be. In 1951, the locomotive would have been a steamer, or an early EMD F3 or Alco PA Diesel streamliner.
Continuity mistake: When Jim Carrey's car goes into the water, it lands upside down. Then he is underwater and gets his coat off. The car is completely submerged at this point. When he comes up for air, the car is only halfway submerged with one of the ends sticking up perpendicular to the surface of the water.
Factual error: If you look carefully you will see that all the cars have license plates with 7 characters on the plates. California did not introduce license plates with 7 characters until the early 1980's.
Other mistake: At the welcome home party, Peter/Luke is at the piano with the female piano teacher. She is helping him remember how to play a song. Problem is, she is playing the keys right in front of her, and he is playing the keys right in front of him (you can tell by their hand placement and where their eyes are looking), but they are both playing in the same octave. His notes should have sounded at least an octave higher. Furthermore, when he starts playing the jazzy sounding tune, the keys we hear are very low and would have been to the left of the teacher. He would have had to reach across her to reach those keys, but instead he is playing the keys right in front of him.