Continuity mistake: When Irene goes to leave the diner after Hank's jokes, his cigarette switches from his left hand, to his right, to his left again.
Visible crew/equipment: When Charlie starts beating the mirror on the train you can see the boom mic on the right of the reflection.
Factual error: The police are informed that Charlie and Irene are 100 miles away. As they board a police helicopter one of the cops tells them they can be there in 15 minutes tops. This means that the helicopter will be traveling at 400 mph. Police helicopters max out at about 130 mph. (01:02:25)
Continuity mistake: In the scene where Charlie shoots the cow, no flame or smoke is seen coming from the gun in some shots, but all the others show flames and/or smoke. (00:30:50)
Continuity mistake: When Charlie and Irene meet with Gerke, Gerke hides his gun under his arm.He then moves his arm to his side but then in the next shot it's back under his arm.
Factual error: Much of this movie was filmed in the Burlington and Middlebury areas in Vermont. Therefore, there are a lot of inconsistencies with vehicle plates and identifying characteristics of things like particular state trooper uniforms/cars/etc. And yes, the abandoned bridge/railroad trestle/etc. is actually the pedestrian bridge over the Otter Creek in downtown Middlebury.
Continuity mistake: Just notice that in the clip at the very, very end of the movie, when Whitey says he thinks he's found the missing thumb, he ducks under without the hose in his mouth. But in the water, it's in. Then when he comes back out, it's out again!
Other mistake: When the cow is seen on the road, note the black and white patterning on its hide. At the end, when you see the cow in a neck brace, it has different patterning. (00:29:50 - 01:44:35)
Character mistake: Throughout the movie, Carrey's Commanding Officer, who is a Colonel, is called Captain.
Character mistake: Shonte Jr. says he can't figure out of the atomic mass of a deuteron. He is supposedly a genius, but this is chemistry 101. Average people might not know how to calculate atomic mass, but a genius would not make a beginning level error like this.