Mr. Alley refers to Nikola Tesla as "Doctor". However, Tesla received his first doctorate in 1908, several years after the events depicted in film take place.
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 designed in the 1960s to protect straight glasses from nicks and cracks when cleaned in automatic dishwashers.
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.
The English "nonik" pint glasses drunk from a couple of times during the film were introduced in the late 1940s. China mugs would have been most common in the era of the film, but that would have made the see-through-the-bottom feature relevant to those parts of the film impossible. A straight sided conical glass would have been possible at the time.
The welded wire mesh embedded in the fence at the laboratory in Colorado Springs was only patented after the events in the movie and would not have been available at the time.
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.
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.
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 Angier puts on his vest he buttons it up while talking to Julia then the shot switches. After it goes back to him he is buttoning up his vest again.
When Christian Bale's character 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.
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.
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.
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.
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.
Angier states that Tesla agrees to see him on February 8th, 1899 in Colorado Springs. Tesla did not move to Colorado Springs until May of 1899.
Borden's plan all along is to have Oliva give his encoded diary to Angier and to get rid of him by sending him to Colorado by providing the keyword. However, this never would have happened unless Angier did something Borden couldn't predict, like kidnap Fallon.
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 is in jail and traps the guard, he begins the scene writing down his tricks. All 10 fingers are visible when two of them should be partially gone.
Near the end of the film as Angier is lying shot on the floor he tells Borden that entering the machine on stage each night was hard, not knowing if he would appear on the balcony or drop in the tank.
Previously, it is shown that the cloned items always appear elsewhere, while the original item remains in the machine. Tesla's experiments with the top hat and the cat, along with Angier's own experiment with himself, all end with the cloned item appearing outside the machine.
While it is logical to assume the clone always appears elsewhere, Tesla clearly doesn't know himself which is the original and which the clone when referring to the hats. It could be that the clone is created in the machine, with the original being transported elsewhere.
Near the end of Cutter"s Summation, Alfred Borden picks up his daughter with left arm. It is very quick camera shot, but all five fingers are revealed.
When Sarah hangs herself in Borden's workshop, you can tell that the rope is attached to the top of her back and not around her neck.