It could be a heater problem, but if the pipes pipes are in a cool location, it could take some time for them to heat up. Water temp heat should be set no higher than 120 deg. If it continues to get too hot, reset the thermostat on the heater, and check its' operation.
You are correct in your description of how a water heater works. We usually get complaints the other way around........running out of water too soon.
You are correct in your description of how a water heater works. We usually get complaints the other way around........running out of water too soon.