How is the third brake light grounded? I once spent nearly two days tracing a problem (with the fuel pump, but the lesson applies) that turned out to be a ground wire that was intermittently failing due to a loose screw and rust at the place where it attached to the body.

Contacts can make a difference; every time we take a trip in our RV, I check the lights, and often have to open up one taillight and look sternly at the contact points, which somehow causes them to work again. But, if you've cleaned and brightened the contact points, that shouldn't be an issue.

Got a multimeter? See if you can trace the current path for a failure point, starting from the hot side of the light fixture/contact points.