Can a thermocouple be partially bad?
The solenoid will require a certain voltage range from the thermocouple in order to operate.  If the thermocouple fails to produce that voltage range, then yes it could be considered partially "broken".  You can probably find the expected output in the service manual and test it with a multi-meter in order to rule this in or out.

It could also be an issue with the position of the thermocouple, a problem with the valve, low pressure or some sort of safety mechanism.  Does your water heater provide any indicator lights to help you diagnose what is going on?