Is it a start capacitor or a run capacitor? What's the type and value / rating.
Edit: Given that I'm off to a nice cool bed...
It it is a start capacitor then it will be a short duty cycle bipolar electrolytic, switched out by a centrifugal switch on the motor armature. Motor run capacitors are Polypropylene film continuously rated ones (with just 3 wires coming out of the motor winding).
Motor capacitors deteriorate over time - they may still show an approximately correct capacitance value but the ESR will have increased to the point where they just can't pass the current anymore. Given the low cost (and apparent desperation!), just change it. If it fixes the problem, you'll be happy. If not then you have a more complex problem that will be harder to find and fix anyway. Play the odds.