Many AC motors have a capacitor that is used to start the motor and then is switched off after it reaches top speed. It is not designed to stay in the circuit for extended time.

You should always start a fan on High and after it's up to speed, switch to Med or Low. If you've been starting in on Med or Low and skipped past the High setting, you never switched out the capacitor and it's probably dead now. Your description is very typical for this kind of problem.

If you read the directions (who reads them for a fan?!?), you'd probably see that they wanted you to start in High and then switch to a lower speed.

It probably isn't worth repairing, but it is worth remembering to always turn it on High before reducing the speed to what you want when you get your next fan.