Continuity mistake: When the Mexican alien is running away, the whole scene is all wrong: He runs for a long time and passes by many bushes and cacti, but when K takes out his gun, the alien is just 10 meters away. Plus, from the POV angle, he passes by the cactus to the left of the guard twice. (00:06:50)
Continuity mistake: In the beginning, when K shoots Mikey, one of the cops gets covered with blue slime all over his face and hat. Watch the right side of the hat: The length of the slime differs between the front and back shots. It's more seen when the scene is about to end and the slime starts to fall down in the front shot, but stays still in the back one. (00:07:25)
Continuity mistake: When K shows the neutralizer at the border patrol guys, he holds it parallel to his face, but in the close-up it's next to where his shoulder meets the arm. (00:07:50)
Other mistake: At the very beginning, when K disintegrates the alien Mikey, an MIB cover-up team arrives on the scene and approaches K at a distance of about ten meters (roughly 30 feet) ; whereupon K instructs them: "Give me a splay burn around the perimeter with holes at forty, sixty and eighty meters from right here! Thank you!" The team members disperse a scant few steps and use their flamethrowers and extinguishers on the cacti and undergrowth in the immediate area for five seconds. However, the distances K describes (forty, sixty, and eighty meters) would require the team laboriously dispersing hundreds of feet into the desert. But five seconds later, they're still only about 30 feet away from K when he says, "That's good! Thank you!" (00:07:55)
Other mistake: The first time the neuraliser is used it shows hours, weeks and years. But at the end of the film the lead character tells J to remember that it's "days, months, years." (00:08:06 - 01:24:52)
Suggested correction: Perhaps there are different models of neuralyzers.
I don't think arguing "perhaps such-and-such" is necessarily strong enough of a reason to correct a mistake. Nothing in the movie indicates they are different models, so it's probably best to assume they're the same.
Visible crew/equipment: In the scene in the beginning where Will Smith chases the alien who jumps off the bridge onto the street. Will Smith jumps from the bridge onto a tour bus, then jumps on the back of a newspaper delivery van. In the passenger side of the van is a cameraman with his back towards the windshield in order to film down the side of the van as Will Smith jumps off the van. (00:10:20)
Continuity mistake: When the truck gets smashed at the start, the tree beside the truck is well alight, but when the man comes out of the house, it is only a little alight. (00:12:40)
Revealing mistake: When Beatrice turns the sugar bowl over to dump the sugar into the glass, a barcode label and handwritten prop inventory number with the name 'Sony' can be seen on the bottom of the bowl. The film was made at Sony Pictures Studios. (00:15:15)
Continuity mistake: The Bug pulls his new "Edgar Suit" skin back tight over his face and the wife faints. In the wide shot, he is still holding it back, but the head is completely normal. (00:15:30)
Continuity mistake: After Kay neuralizes Jay the first time, Jay 'wakes' up in a restaurant and Kay hands Jay a card. He tells him to report to that location. The address can be seen written on the back on the bottom edge of the card. Jay flips the card over to read the address and it's still on the bottom edge of the card. When flipped, the address should now be on the top edge of the card. (00:22:00)
Continuity mistake: In the beginning of the film when "Edgar" kills the exterminator, the exterminator's truck has a small almost unnoticeable plastic roach decoration on top. Yet during the rest of the movie the roach is huge and it sticks up away from the vehicle. (00:22:30 - 00:33:20)
Continuity mistake: When Will Smith and the other recruits are taking their written tests, the white table first appears to be situated between the second and third men from the left. When Smith goes to the table to pull it across the room, it is much further to the right of the scene. (00:24:25 - 00:26:35)
Continuity mistake: The MIB applicants are all sitting in the testing room and Zed talks about how they are all the best of the best of the best like, "Rangers, Navy SEALs, NYPD". The only sailor sitting in the room is not a SEAL, he's EOD. You can tell because he is not wearing the Special Warfare badge on his uniform, he's wearing the badge for Explosive Ordnance Disposal. (00:24:35 - 00:27:30)
Continuity mistake: Throughout the whole movie it is summer. Green trees with leaves, green grass, etc. After the scene where Will Smith is deciding to become a Man in Black (summer), there is a shot of the Bug Mobile driven by Edgar where it is obviously winter. A wide shot of Manhattan with steam coming out of the tops of some of the buildings, and some trees without leaves in the background (winter). Then in the next scene and those following it is back to summer. (00:31:20)
Continuity mistake: After doing his recruitment test, Will Smith and Tommy Lee Jones are sitting down on a bench and talking. The distance between them decreases between the first and second shot. (00:31:20)
Audio problem: When Jay goes to join MIB, he greets the security guard with "'sup pops?" without moving his lips. (00:33:40)
Visible crew/equipment: When J puts the glasses on for the first time, a lighting screen is reflected on the right lens. (00:38:30)
Continuity mistake: When the jeweller leaves his shop, the shop window clearly shows the reflection of the parked exterminator's van. Then you see the van being driven up the cross-street as the jeweller leaves in the taxi. Then Edgar/bug parks the van, goes to the shop, and you can see the same reflection in the jewellery shop window. (00:38:35)
Continuity mistake: When Zed sends Kay and Jay out on Kay's first mission, Reggie's unauthorised departure from Manhattan, Tommy Lee Jones legs cross and uncross from shot to shot. (00:39:25)
Continuity mistake: When J and K first get in the car (the Ford POS),the windows keep switching from up to down. Interior shots Will Smith's window is up, exterior shots his window is down. (00:40:30)
