Its possible that the Tesla simply won't charge at all if the current or voltage on the line becomes too low. In this case, it would be "plugged in" but not charging.
Edit: Lets figure it out: Range=244mi. Battery capacity=58kWh. So: 5 miles takes 1.086kWh. 10% is lost in chemical conversion so we really need 1.207kWh for those miles. We pull at a rate of 1.8kWh/h from the plug. So 600 watts or so is lost elsewhere. We know the "always on" battery cooling system alone takes around 150 watts. Its not that far fetched.
Edit: I deleted my comment below where I calculated the cost of keeping the car on standby because I don't think I made clear enough (and didn't want to type it all twice) the difference between the actual discharge rate of the battery and the amount of power required at the charger to stop it. It just seems to take a lot of power at the charge port before any gets to the battery. The battery seems to discharge at an average rate of just 30 watts, but it seems to take a much, much greater amount of power input to prevent this and failing to provide this power has dire consequences. Why?