Continuity mistake: When J and K fall off the roof holding onto a door, they land opposite to each other. A frame later, from a different angle, they're laying next to each other. (00:17:55)
Continuity mistake: When J steals the blue car keys, a nerdy looking guy with an orange and white striped shirt and glasses is walking towards the hotel. After J drives zooming through New York, the same guy is seen crossing the street walking the opposite way. (00:39:15)
Continuity mistake: When J exits the Flatiron building in 1960 a woman walks right behind him, then the shot cuts to his POV of a group of hippies, then back to the previous shot of J and the woman obviously continues her walk. She should've been several meters ahead already. (00:38:30)
Continuity mistake: After K is falling with J from the rooftop, big chunks of concrete start to fall, yet disappear in some shots. When they land, the ground is clean, yet in one of the close-ups of K a chunk magically reappears next to his head. (00:17:50)
Continuity mistake: In the Headquarters in the altered time line, when J is crying out that he and K spent 14 years of stakeouts, cowboy music, etc. A woman in the back swaps from standing by a screen to walking 20 feet to the right, between shots. (00:28:40)
Continuity mistake: When K tells J he's suspended after asking about Boris, they get into a brief confrontation about it. As K is storming away to take the car, leaving J back at Wu's without a ride, J's suit is still dirty, wet, and nasty from his fight with the alien fish. Moments later, J is seen going to MiB headquarters and now his suit is clean and in perfect order. (00:19:45)
Continuity mistake: As J is being put in the "big ass neuralyzer", they first have him in front of it strapped in a metal bracket. As they begin turning him upside down, the white suit guy on the right reaches over and begins opening the hatch on the neuralyzer. You see him turn the handle and hear the creak and see it slowly crack open for a moment before the shot cuts. Then two shots later as the white suit on the left is about to put the bite guard in J's mouth, the hatch is suddenly closed and the guy on the right is reaching over to open it again. (00:46:10)
Continuity mistake: When Jeff is talking to J on the Flatiron, he is looking at J except for a brief shot where strangely he is suddenly facing left without having turned his head. Then a frame later, from another shot, he is back facing J. (00:33:50)
Continuity mistake: Before leaving in the jet packs to go to Cape Canaveral, O steps off K's jet pack twice between shots. (01:19:25)
Continuity mistake: On the Apollo docking bay, just as J says "You're gonna lose", Boris has J pinned up against the scaffolding. Then as Boris says "Let's agree to disagree", J is suddenly not pinned against the scaffolding, leaning forward. J also goes from looking ahead to looking down. (01:28:35)
Continuity mistake: When Griffin shows J and K the vision of the baseball game, J walks back and then walks back over to Griffin. Griffin then places his arm around J and his hand ends up around the top of his arm. In the next shot, Griffin's hand is suddenly on J's shoulder. (01:07:05)
Continuity mistake: When J and K are talking to Agent W backstage, as Agent W says "You're out of your mind...", he is looking straight forward at K, head remaining still. Next shot, he is looking at K with his head is suddenly shifted to the left. (00:56:05)
Continuity mistake: As J and O are talking about temporal rifts, after J states Boris was on Lunar-Max, O says "If Boris could figure out how to time jump, he could seriously change the course of history." In this shot, her left hand is raised and her fourth finger is shown to be placed on her thumb. Next shot, her finger and thumb are no longer touching and her finger is now pointing to the right. (00:30:30)
Continuity mistake: When J believes K is playing a joke on him, as O says "who is he to you?", she raises her finger. In the next shot, the position of her finger changes. (00:27:25)
Continuity mistake: After Z's memorial has ended, J and K are driving. As J says "Because I was too tight", he is looking at K, doing a mime for tight with his right hand as the line ends. Next shot, he's looking straight forward, not doing the mime anymore. (00:10:15)
Continuity mistake: In the car after Z's memorial, K asks J, "You lose something over there, Hondo?" As he finishes this line, J is looking at K, sitting up. In the next shot, J is looking at K with his back resting against the car seat. (00:09:50)
Continuity mistake: After Boris has blasted a hole in the steel wall of Lunar-Max, Boris grabs onto his girlfriend's arm who is getting sucked into the vacuum of space. The first shot shows his hand wrapped around her wrist. A few shots after, his hand is further up her arm. (00:05:10)
Continuity mistake: Boris' girlfriend is first shown to have a tattoo with "Boris" over her back. A few minutes later, after Boris says "I'm a bit of a romantic", the Lunar-Max guards walks over to the cake in Boris' girlfriend's hands and the tattoo on her back is shorter. (00:00:35 - 00:03:00)
Continuity mistake: At the Apollo rocket launch platform, J calls Boris the Animal twice, who lifts J up and launches him away. J glances at K, who's got both arms stretched out trying to save himself. A shot later, in the close-up, only his right arm is stretched out. (01:28:40)
Continuity mistake: When the little alien sneaks inside the Boris' hand, the fingers are stretched out in one shot, yet bent a frame later. (00:03:50)
Answer: Earlier in the movie, the way K was talking about Boris was different than what J had come to expect of his partner. So J wanted to know what happened between Boris and K at Cape Canaveral that changed him. So more of what did Boris do to K rather than just the report of what K did.