The Simpsons drive to the Academy in Homer's pink sedan. But when Homer, Marge and Maggie leave, it is in Marge's station wagon, even though it is on the same day.
After Bart's megaphone mayhem, just before Chief Wiggum discusses Bart's punishment, you see Homer's classic pink sedan in the driveway and a police car, neither with the windows smashed.
When Lisa is talking to Grampa on the phone, the phone switches between having nine buttons and twelve buttons.
When the family observes the poetry class at the military academy, the teacher's appearance changes, with his hair being black in one shot and gray in the other.
Near the end, when Bart bumps into the cadets who snuck out at night, the brown-haired cadet in a red jacket and the blonde cadet in a green jacket switch places between shots.
It has been proved that Bart's megaphone prank would not have worked, apart from causing hearing damage due to all the acoustic feedback from the microphone and speakers.
Lisa's sub-machine gun is shown to have just a small clip, so there is no way she could have fired many dozens of shots before it finally ran out of ammo.
The educational film THE MOON OF EARTH, supposedly made in 1952 during Adlai Stevenson's presidential run, contains a prominent American flag with 50 stars (post-1959) instead of the period 48, shown behind Governor Stevenson. Another flag, shown later, is correct.
When Bart is shooting the six-shooter grenade-launcher, we clearly hear six shots, and yet afterwards, the drill sergeant says that Bart missed only one out of five targets.