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.
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.