You do not state whether the motor is driving a mechanical load during your tests; if it is then the load may just be too great to permit motor rotation.
If the motor is mechanically disconnected from a load, then I feel there is something not quite right with either your motor or PWM drive system.
At all values greater than 0% there is FULL system voltage being pulsed to the motor. It is the duty cycle (ON time) of the PWM that determines the mean voltage applied to the motor, and hence the motor speed. So, even considering motor friction, anything greater than say 5% should be capable of driving a mechanically disconnected motor.