In one of the earlier scenes of the movie, Alfred and Robert are seen in the dressing room after one of their performances together. Robert puts on a brown vest and crosses the room, and in the next shot he is seen pulling it on over his shoulders once again.
When Borden writes a note to leave for Angier by the cemetery, he clearly makes a period to indicate that he is done writing. Later when Angier reads the note it only says "Tesla", without a period.
When Borden and Olivia are speaking in Borden's workshop, Borden keeps switching between shots from leaning back with his arms crossed to leaning forward with his hands on the table.
After Root gives the "show" he takes out a drink while Angier is walking up the stairs, then he brings the drink to his side and as the camera angle switches he brings up two hands without the drink.
As Angier is walking towards the box in Tesla's empty room, you notice the hotel manager deliberately bring his hands from behind his back to his front. In the next shot, taken from behind, his hands are behind him again.
Angier states that Tesla agrees to see him on February 8, 1899 in Colorado Springs. Nikola Tesla did not move to Colorado Springs until May of 1899.
When explaining the underwater escape trick to the judge, Cutter demonstrates that the padlock is a dummy and easily unlocked with one hand, yet when Angier's wife is drowning he chooses the long and arduous route of attempting to smash the glass with an axe rather than simply unlocking the padlock. However, many times in the movie it is made clear that the 'secret' of an illusion should never be revealed to the audience - perhaps not even, as in this case, when a life is at risk.
When Angier visits Tesla in February, it is obviously winter, with snow on the ground. Yet after a brief meeting they venture out to a balcony, where it is summer, with green foliage, and no breath visible.
When Angier and Root, both played by Hugh Jackman, are in a circular tracking shot together, Angier wobbles up and down slightly, revealing him to be the inserted visual effect.
When Borden produces the key to their new house to his family he clearly has eight intact fingers. Yet, much is made of the damage to them in the previous scenes.
Mr. Alley refers to "Doctor Tesla". Nikola Tesla received his first doctorate in 1908, several years after the film takes place.
Borden mentions that a "penny" could be slipped into the barrel of the pistol used for the bullet catch trick. English pennies of this period were substantial and wide, much wider than the barrel of the pistol being used by Borden could accommodate.
The spider web shaped cracks that appear in the glass boxes when characters attempt to smash them indicate they are made of laminated safety glass, a material held intact by a thin layer of flexible vinyl sandwiched between two sheets of glass. Laminated safety glass was discovered by accident in 1903 and patented in 1909, well after the time line of the movie.
When Angier and Cutter discover playing cards attached to the bottom of their pint glasses, the glasses they are drinking from are 'Nonic' glasses. These were not created until 1914, in part to protect straight glasses from being chipped if knocked over.
During the magic show where Alfred meets Sarah for the first time, on both sides of the stage there are statues looking exactly like the two guardian statues found in King Tutankhamen's tomb, not discovered until 1922.
When Angier gives the first show of his last 100, he explains the risks of the underwater-unleashing trick. The camera switches between shots of him and Borden, and we see the crowd behind Borden (soundlessly) clapping although Angier is not done explaining.
When Root is tied up above the stage, you can see what seems to be a security wire that comes out horizontally from both sides of the stage and is attached to Root as he is being lowered down.
After Julia drowns, anyone could investigate the knot tied, because obviously she couldn't untie herself. Therefore Angier should already have known which knot was tied.
Borden is sent to jail and receives the death penalty for murdering Angier. It is questionable Borden would be convicted for that. Angier dies during a magic trick - to which Borden is only an observant - by falling into a water tank - that Borden did not place there - and drowns. The only thing Borden could have been accused of is that he made no attempts to rescue Angier.
Angier says in narration, "Borden writes as if no one but he understood the nature of magic." In this usage, "but" is a preposition meaning "except for" and takes an object: Angier should say "no one but him".