In the airport scene at the end many of the troops are armed with M-14 and M-16 rifles (as indicated by the flash suppressors.) While these weapons would be contemporary with the filming, they are much later than the film's time frame (late 1950s).
In a voice over near the Montana - South Dakota border, Holly says "...at the very edge of the horizon we could make out the gas fires of the refinery at Missoula, while to the south we could see the lights of Cheyenne, a city bigger and grander than I'd ever seen." From anywhere along the border, Missoula would be over 600 miles to the west, and Cheyenne would be some 300 miles to the south; nothing short of a satellite view would allow you to see both at the same time.
In the scene where Kit records his confession on a record, he places coins in the machine and then flips the switch from 78 rpm to 45 rpm. The instructions above the switch clearly says "Before Inserting Coins Select Recording Speed." If he were following directions he would have flipped the switch then inserted the coins.
At the beginning of the movie, Kit meets Holly after a long day as a garbage man and his white t-shirt is spotless. Kit then asks Holly to take a walk with him, at which point the t-shirt is magically dirty!
At the end,when Kit surrenders he raises his hands straight up at first, but three seconds later they are slightly risen and bended over his head. After another two seconds they are back to their initial position. During this sequence the distance between him and the Cadillac changes as well.