Continuity mistake: When J is charging Boris and keeping track of the spikes that the alien shoots at him so he can dodge them on the second try, Boris first shoots and hits J in the left shoulder. Then the second shot hits J in the right as J says "right" and the third shot hits his left side. Then when he goes back in time a few seconds, the only thing to change is J knows the sequence of which Boris will shoot the spikes. Only it is different. J dodges the first one and says "right" as it passes his right side. Followed by two lefts, and then a forth one on the right. Yet previously, it was left first, then right, and then left again only three. It goes from left right left, to right left left right.
Continuity mistake: As the hand creature busts the shackles off Boris' legs in Lunar Max, you can see the bottoms of his pants over his metal prison boots. There are only a few folds and wrinkles over his left foot, and the pants over right foot are hardly wrinkled at all, or affected as the creature climbs up his right leg. But only a couple of seconds later, we see another shot of Boris' feet as the creature breaks off the metal sleeve of his arm, and suddenly his pants legs are very wrinkled. Boris was standing perfectly still the entire time. Boris' legs are also suddenly closer together.
Plot hole: Lunar-Max prison is obviously part of MiB, them creating it for the purpose of holding Boris the Animal, and other criminals of worth as well. A high tech prison built on the moon with cutting edge technology and armed guards out the boot. Boris trashes the place, sounding all kinds of alarms and busting out, even blowing a sizable hole in the side of one of the complexes. He then jacks a ship off the moon's surface and crashes it back down to Earth. MiB doesn't know about it until almost a full day later when they realise the ship they found was from the moon, and Boris had escaped. It makes no sense that it took them that long to find out the information. A maximum prison, especially one to contain aliens on the moon, would be in contact with MiB. And any alarm that goes off there, especially for a break out of a high danger prisoner such as Boris the Animal, would have sent an alert to MiB headquarters right off if there was any logic to their system at all. There is only about a 1.5 second delay for a message traveling at light speed to reach Earth from the moon due to its distance. So within seconds of Boris busting out, MiB should have already been mobilizing to track where he would be going and intercept. There's no way, that logically, Boris could have escaped, made it back to Earth, and been there for nigh a day without MiB knowing about it and having their agents already tracking him down.
Factual error: When J goes to the electronic store to get the time jump piece from Jeffrey Price, all of the TVs in the store are showing "Wheel of Fortune", which in NYC is shown on ABC TV channel 7. But when the show is interrupted for the emergency story about the invasion, the TVs show NBC TV Breaking News, which is channel 4 in NYC.
Continuity mistake: After busting out of the cake and killing the two Lunar Max guards, Boris' hand creature goes to free him. It first blasts spikes at the shackles on his legs and busts them, then begins climbing up to work on Boris' arm piece. In the shot where his leg shackles come off, you see the spike the creature shoots at them stuck in the metal and hanging off to the side. But in the shot where his arm piece falls to the floor, these spikes have vanished from his leg shackles.
Continuity mistake: In the first Men in Black, when K is deleting J's history; He deletes his birthday which was in 1975. The third film is set in 1969 and when K meets a young J, he is about 7 or 8. This would mean he was born no later than 1962, not 1975.
Continuity mistake: When J is digging through files to try and find out what happened between Boris and K, O comes in and talks to him for a moment. As she is leaving, shots of J shows he is leaning back in his seat. But when O is shown, the glass wall she is by shows his reflection, leaning forward and slouching.
Continuity mistake: When the racist cops are frisking J, one pulls out J's gun and questions him about it. In the reverse angles, the cop is holding the gun over J's shoulder with almost all the handle in his hand. In the shots showing J's face, the cop is holding the handle so that some of it is sticking down, and it is in front of J's shoulder.
Factual error: When J looks through the log book telling when Boris was sent back in time, one of the previous dates says April 31st. There are only 30 days in April.
Continuity mistake: As J pops his head up on the tall building, there are windows immediately behind on his right and left and both are closed. The camera pulls up and back to show the scale of the building, then cuts back down to J. Now suddenly the window on his right side is open, and the lamp in the room is on. The window also changes from closed to open a few times in this scene.
Factual error: In the film, in 1969, prior to the launch of Apollo 11 the moon is shown to be a full moon, however, in reality, on the night before the launch the moon was a tiny crescent moon.
Continuity mistake: When the huge alien fish throws J through the window of Wu's, J knocks over several rows of frogs hanging in the window as this happens. As he finishes being thrown out of the window, only one row of frogs is left. A few minutes later as K suspends J, K goes to exit Wu's and now there are suddenly two rows.
Continuity mistake: As soon as Boris blasts the door down, there are guards on the grounds and guards behind some of the cells in the background. After Boris blasts the the steel wall, the guards on the grounds are sucked into the vacuum of space, but the guards hiding in the cells in the background have vanished completely. They certainly weren't sucked into the vacuum of space because they were protected by both the cell bars and the ceiling above them.