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.
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 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.
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.
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.
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 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.
The goof items below may give away important plot points.
Crew or equipment visible
When Root is tied up above the stage after Borden sabotaged Angier's 'The Real Transported Man' trick, 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.