Plot hole: Julia dies in the underwater escape trick tank because the axe couldn't get her out in time. However, if she was able to open the lid from inside the tank, get up, over, and down, then close the lid (all quickly, silently, and without assistance) when the trick goes properly, then there is no reason that the four men standing around could not have simply lifted the lid, either acting a key in the fake padlock to preserve the trick, or using a key if it's real. At this point she could have gotten her head above water and draped her hands over the side of the tank to stay up. That done, the crisis is over, and she must only wait to calm down enough to get herself down, or be assisted. Cutter built the tank, so he knew how strong the glass was. The only explanation for using the axe was to allow Julia's death, thus advancing the plot.
Plot hole: When Angier and Cutter show the machine to Ackerman in the abandoned theater, it is shown that, after Angier enters the machine, he completely disappears. No duplicate is left behind. Since it is not a completed scenario, there is no trap door for a duplicate to fall in. He is seen disappearing, leaving none behind, and appearing at the other side of the theater to Ackerman's surprise. However, in the end of the movie it is revealed that this machine never disappeared anybody, it just created duplicates at a certain distance. So why didn't a duplicate remain in the machine when they showed it to Ackerman? And if the reason is that they could fix it, so a duplicate wouldn't remain, then why would they still create duplicates for the rest of the film? This little scene breaks all the logic of the film.
Other mistake: This mistake occurs when Borden crashes Angier's act. Borden emerges from the trick door for the Transported Man trick, feigns surprise, and then "discovers" the bound and gagged double suspended above the stage. When Borden says "Ah, the Great Danton" you can see the ring and pinkie fingers on his left hand moving, even though he had already lost them by this time in the movie. We know that it has to be after the time BOTH twins lost their fingers because Olivia observed Borden's trick and says that "it is the same man" because she was fooled by both Bordens having their gloves stuffed.