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: It is totally unlikely that a museum as careful and important as the Smithsonian would leave the crates un-nailed, which could make the pieces inside get damaged. Of course if the crates weren't open, none of the figures could come up to life.