When Cooper breaks into the Admin building to steal the student records, he approaches a filing cabinet and tries to open it and says it's locked. However, the lock on the filing cabinet at the top is ejected, which is an indication that the file cabinet drawers are unlocked. If the cabinet was truly locked, the lock would have been pushed in. However, this could be deliberate - Cooper isn't exactly the brightest bulb; it's possible he just doesn't know how to properly open a file cabinet.
Buckley mentions losing half his brain to "falling asleep at Newark Airport", but if he'd really lost half his brain, he'd at least be in a stroke-like state, unable to control half his body. However, Buckley is paranoid, so this is excusable.
When Cooper and Josh are breaking into the library at night, Cooper wraps the rope around a support beam. When Josh takes up the slack to lower Cooper into the library, the rope he is holding dangles loosely down, revealing it to not be taking any weight and also showing it's not wrapped around the beam. He also doesn't move at all when Cooper drops in and supposedly puts about 180 pounds of weight on the other end of the rope that Josh is holding.