could it be a bad water pump?

Only if pump is leaking. Any fluid dripping when running? If not check oil to see if water leaking into engine, oil will look creamy.

The pump can be bad without leaking. Most times a pump will leak if bad unless its the impeller. You will see coolant leaking if this is the case now that you know about it. I would check all other possible places for leaks too, it may not be the pump. Have seen any leaking? Also Have you noticed a antifreeze smell or a sweet smell around the car? Check you oil, does it look milky? Is there steam or white smoke coming from the tail pipe? Check for leaks around the thermostat. Is leaking somewhere somehow. Locate the leak. Hate to say it but you can not rule out a head gasket even if this was not the leak now that it ran warm or almost hot it could warp the head and blow a gasket.