Deliberate mistake: When MIB agents exit the lift to get into the main hall of the MIB Office, the lift opens halfway up the wall and they need to go down a little platform. But when K and J return during the lockdown and face the trashcan with guns, the lift opens at floor level, so it can be conveniently peppered with bullets.
Continuity mistake: In the scene at the lockers in Grand Central Station, Agent Kay takes his watch, an illuminated Hamilton digital, back from the little animals living there. In one shot, Agent Jay puts an analogue watch in the locker as a replacement, but in the next shot it is the same old Hamilton digital owned by Agent K. (This is mentioned in the DVD Audio Commentary with Barry Sonnenfeld).
Deliberate mistake: The opening of the film says it takes place in July 2002, but when we first see Will Smith you can see the World Trade Center in the background which had been destroyed ten months before. [The film was made before the 9/11 attacks, and the towers were left in as a sign of respect - still a mistake, but there's the reason.]
Deliberate mistake: When the trashcan opens fire on the lift with Jay & Kay in, the distance and angles mean it would, at most, be able to hit up to the edges of the rear doors. When the inside of the lift is shown, the bullets have somehow managed to reach right to the edges of the lift.
Continuity mistake: After Jay and Kay go to see Jeebs about the deneuralizer, Kay leaves and the bad aliens come crashing through the walls. Jay shoots Jeebs head off and, as it's growing back, he says he got shot "right in the pie-hole" (mouth). However he was facing away from Jay when he was shot and it was above the mouth area.
Continuity mistake: When Jay and Kay make their visit to Jeebs, Jay is talking to Jeebs and you see Jeebs making a statement about how his business is 'banging' and when he see Kay walk in, he points both his hands at him. The camera angle changes and Jeebes is now seen with only one hand pointing at Kay.