Deliberate mistake: After Wolverine arrives at 3 Mile Island he approaches through a thick fog. Above him are a series of conveyor belts that are used at a coal power plant, not a nuclear plant such as 3 Mile Island. The belts are unique to coal mines and coal power plants, and would not be found at a nuclear power plant. The filmmakers obviously couldn't film on the site of a nuclear plant, so filmed on a coal plant to fill in.
Deliberate mistake: After being arrested, Breaker, who is Northern African, a region where French is the second if not first native language and speaks with a French accent throughout the movie, pleads with the French authorities in English. (01:10:30)
Deliberate mistake: When Selfridge says "Pull the plug" after Hometree is destroyed, two soldiers, one male and one female, go to shut off Jake and Grace's avatar links. The female soldier goes straight to Grace's link and terminates it, but the male soldier gets delayed (punched by Norm) before he can terminate Jake's link. This is deliberately shown out of sequence - we see Jake's link terminated before Grace's. The result is that we see Grace's avatar fall unconscious when Jake's link is terminated, and Jake's avatar fall unconscious when Grace's link is terminated.
Deliberate mistake: Though the film takes place over the course of several weeks, at night there is always a full moon.
Deliberate mistake: In the scene where we see Bella in her room looking out the window, the months are passing, but she is always wearing the same clothes. (00:28:30)
Deliberate mistake: When Sam is watching the security footage of his predecessors being disintegrated in the chamber, the disintegration happens the instant the coffin door is shut. However when he goes in to test it himself there is a ~60 second wait as a congratulatory video plays before the disintegration occurs. (01:00:00)
Deliberate mistake: A branding iron loses its bright orange glow relatively fast when it's out of the fire. When the police enters the room it is still very bright orange even though it has been on the floor for some time.
Deliberate mistake: In the link between the movie reel and nightclub scenes a train goes past. While the sound effect is convincing, the video has been reversed. The headlights on the train give it away, but cars under the bridge are also seen in reverse. (00:27:55)
Deliberate mistake: A flare thrown on top of a snow-covered mound of logs/twigs is not able to engulf the area as fast as it does. (01:13:13)