Deliberate mistake: Towards the end of the movie fish are seen to swim down in order to submerge the net in which they have been caught. Although the attempt seems logical, it actually needs power against the net to push it down. None of the fish seem to touch the net at the bottom however and if it is the whole mass of fish achieving it, then the bottom fish should have been squashed against the net which (again) isn't the case. 'What brings the net down?' is the question here. (Done deliberately by Disney/Pixar to be visually aesthetically pleasing.)
Continuity mistake: In the sequence with the boat, its position changes. When Nemo is looking at it with his friends, it is a long way away. When Nemo gets angry and swims to it, it's suddenly very near the reef. Then, the diver catches him when he's only been swimming away from the boat a few seconds, but the diver has to swim a long way back to the boat. Finally, when Marlin follows the boat, it's very near again, because he gets there before it starts, even though he was temporarily blinded by the camera. The boat does all this without raising the anchor until it leaves.
Gurgle: Oh, the human mouth is a disgusting place.
Seagulls: Mine. Mine. Mine. Mine.
Nigel: Oh would you just shut up? You're rats with wings.