[wdfifteen]

Start capacitor is a common culprit. The start capacitor stores energy when the motor is running. When the motor starts again, the capacitor unloads, giving the motor extra current to get going. If it fails the full start current has to come through the fuse. So yes, a bad one can contribute a lot to the load on the fuse.
Long term high current can make the link in the fuse sag over time and fail (especially if it's mounted horizontally), so that is a possibility. You can replace the fuse, but the high current issue is still there.