Thanks Runs with Bison. Since I posted this I asked a buddy of mine who is an E.E. and also a pretty good electrician. Here's what he had to say:
- the capacitor is the wrong size
- the voltage is not correct (especially if it's too high, but could also be if it's too low)
- it's taking too long for the motor to get up to speed, or speed sensor (if there is one - sometimes a centrifugal switch) doesn't release the capacitor when the motor is up to speed.
- any kind of drag on the motor (bad bearings, fan rubbing the housing)
- a fault (partial short) in the windings of the motor
- a partial ground anywhere in the motor/capacitor circuit
- a bad connection, probably on the capacitor
There are a few other possibilities, and most of these can be checked out. It's even possible (though not likely) that the "new" one you put in was bad. But if you've replaced it more than once and it's still happening, that's VERY unlikely.