Deliberate mistake: Leaven spends too much time thinking about if 645 and 372 are prime numbers to be believable. But she knows the factors of 649 and the result of 26 to the power of 3 almost immediately. Plus anyone with any maths knowledge whatsoever would know that any even number (or one divisible by 5) can't be prime, so 645 and 372 could be dismissed without even thinking about it.
Audio problem: When Will Smith is looking at the two space ships at the MIB headquarters, if you look in the background Zed and K are there typing on the computer. But if you look closer you can tell that they're not even touching the keyboard and the sounds are just added into the movie. (01:16:00)
Continuity mistake: Toward the end, Carmen Ibanez gets stabbed through the shoulder when she is brought before the big master alien thing. If you were to sustain such an injury, there would be so much swelling, not to mention just destroyed tissue, that you wouldn't be able to move that arm for weeks. Yet a few minutes later, she is seen hefting around a very large assault rifle, and seems to have no difficulty. I guess it's a pretty standard action movie "heroes can temporarily ignore injury in order to save the day" kind of oversight, but it just seems exaggerated in this case, considering how severe the shoulder trauma looked. (01:48:15 - 01:53:40)
Revealing mistake: Right after they "create" Leeloo and she is lying in the plastic cage with the little straps on her, the guy approaches her and says, "if you want out, you are going to have to work on your communication skills." She gives him a dirty look, and then you can see a huge hole, pre-cut in the glass, that she busts her hand through. (00:29:47)
Continuity mistake: Costume continuity throughout the whole film. When the female scientist gets changed into khaki pants they are tight fitting, they alternate between tight and baggy throughout.
Continuity mistake: In the middle of the hotel fight when Toby and Deliverance go through the door in their hotel room you see the door open and a number of shots impact it. The characters then go through the second of the double doors. This is commented on in the DVD commentary. Not only do the bullets not go through the second door, they don't go through the first. On top of that, you see a shot with the two doors opened, but when the bad guys enter the room the double doors miraculously turn into just one! This layout is then used for the rest of the scene.