Hi Tim & Welcome!
Well, it could be a number of different things and you are probably in for some investigation and electrical testing but I'm going to go out on a limb and say that they are probably both in the up position on purpose.
When I got my '60 one of the back windows was in the up position and wouldn't go down and I found the motor disconnected and the window stop inverted to prevent the window from moving down. One of the tracks was a little bent up but I was able to straighten it out and reconnected the motor and adjusted the stop and the window has worked fine ever since. I'm guessing that the window maybe came off the track or somehow became a PITA for the PO who "fixed" it so it wouldn't open and cause any more grief.
There are also circuit breakers for each motor and grounding can sometimes be problematic. My driver's side door window started acting up a few years back and I found that it didn't have a good ground at the motor.
Good luck !
1960 HT 352
TBird Registry 34042