That sounds about right to me. If the capacitor is good then the switch might not be making contact. The way it works is when the motor first starts, the capacitor is connected to it, through the switch. As the motor gets closer to full speed, the centrifugal part of the switch, pulls away the connection and removes the capacitor from the motor. If it is stuck or dirty, it does not make good enough contact to get the motor moving. So it just sits and hums, eventually the internal thermo switch kicks in and shuts it down so it can cool off.
If you need more specific help on how to check or fix it, please take and post some pics of the back of the motor with the cover off. That way we can see which style and where to look for the issue.