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.)
Deliberate mistake: In several scenes, Sky Captain chases the enemy through the streets of New York. At even the slowest speeds to maintain flight, the hairpin turns around street corners would generate forces equivalent to 15 to 20 times the force of gravity. Even if the aircraft could survive these forces, the pilot and passenger could not.