Continuity mistake: A montage scene of newspaper headlines early on has March 12, 1939 as the date. Much later in the film, Jude Law asks Gwyneth Paltrow the date so he can calculate the position of a star. She responds "March 2" but there is absolutely nothing to indicate that nearly a year has passed.
Continuity mistake: When the robots invade New York City and are advancing on Polly in the street, the camera pans down to the robots' stomping feet. There is nothing there but open space for them to tromp on. The camera pans back to Polly taking a picture, then back to the robots' feet. Suddenly, a car has magically appeared in the path of the stomping robots and gets kicked towards her.
Factual error: At the very end, Polly Perkins takes a spectacular picture, using the very last shot on her only roll of film. When Sky Captain tells her she left the lens cap on, she is mortified. But she would certainly know that on her Argus 35mm camera, all she had to do was remove the lens cap and retake the shot. Unlike modern idiot-proof cameras, hers has no mechanism to prevent double-exposures.
Factual error: After the robots first attack New York City, a series of newspapers is shown. The German one is full of mistakes. It says "Sehr grosse metallic Maschinen stehlen stehlen Reserven." The word 'metallic' would be 'metallisch' in German and they would write robots as 'roboter'. 'Stehlen' (steal) is written twice.
Continuity mistake: When Polly stands up while Joe is attacking the robots, she is on the corner of a sidewalk. A few shots later, when Joe flies over her and she says "Joe", she is at a different place on the sidewalk. She was walking really slowly, so it's impossible that she moved that fast.
Add timeDr Wilson
Deliberate mistake: Just before Sky Captain goes to the rocket ship he punches Polly in the face hard enough to knock her out, and yet she has no swelling nor any type of bruise for the remainder of the movie. A punch that hard would have surely left at least some type of mark on her, especially with her being so fair-skinned. (Yes, this was deliberately done, hence a deliberate mistake.)