I've always loved the Men in Black films, but I must admit that there are a number of plot holes.
1 In the third film, Boris goes back in time to kill Kay before he could have stopped the Boglodite invasion in 1969. This was also long before Kay hired Jay, and yet, Jay still works for MiB. Surely if it were not for Kay, Jay would not have been hired.
This is an example of Bootstrap Paradox. It's a common time paradox. For explanation you can visit my list,
"Top Ten Scientific and Philosophical Paradoxes". - Kiteretsunu
2 In the third film, time travel is shown to be possible. If this is so, why could it not have been used in the last two films, or used to travel to a point in time where, in Boris' case, killing Kay, or in Jay's case, saving him, was easier?
3 In the third film, Kay's objective in the climax is to attach the ArcNet to Apollo 11 in order to shield Earth from the Broglodites. In which case, would it not be easier to ask NASA to put it there instead of bypassing the guards and climbing the rocket?
4 In the first two films, we see a long list of aliens which came to Earth primarily with the purpose to invade it. But in the third, we establish that the Earth had to be shielded from another group of aliens. How does the ArcNet allow some in, not others?
5 In the second film, the Light of Zartha is kept safe from Serleena, and was kept on Earth. Had they kept it elsewhere, would Earth be under no risk, as the fight over it would not take place there?
Think about it. In the third film, they have an alien prison on the moon that has been in operation since long before the second film. The intention behind that was to seal anything potentially harmful away from Earth. Thinking what I'm thinking? - PositronWildhawk
6 Throughout the series, MiB neuralyse people to keep their business safe from others, and was also used to erase the memories of retiring agents. How can this constantly be timed to the nearest second?
7 In the third film, Boris escapes from a prison on the moon and subsequently invades Earth. Without MiB knowing about it. Surely, for maximum security, one would have a system in which MiB receive a signal should there be a breach.
Actually they did know about it.
8 In the third film, we establish that a maximum security prison exists on the moon. The prison in question holds Boris captive since he was arrested in 1969, at the time that man first landed on the moon. How could they afford the time to build it?
9 In the third film, we establish that a maximum security prison exists on the moon. This place was kept secret from everyone else, despite the fact that anyone would be able to see it with a telescope.
10 In the second film, MiB's main base is attacked and everyone there is kept hostage. Would it not occur to MiB to have a backup plan for this situation? Perhaps a base for reinforcements and rescue parties?
It seems somewhat bizarre that an organisation for interstellar security has a more domestic security problem. - PositronWildhawk
11 In the third film, Agent O is being portrayed as if she's been a member of The MIB at least since 1969 having known and worked with Zed and even having a personal relationship with Kay. However, she is nowhere to be seen or even mentioned in the first 2
12 In the third film, Kay has neuralyzed young Jay in 1969. While young Jay was neuralyzed, Kay was clearly giving Young Jay a positive memory of his dad yet in current times Jay is in the bar ripping his dad in front of Kay.
13 Kay's love life.