Modern charging and power conversion systems are designed to operate under specific use cases. When the Tesla charger first sees line voltage, it likely does some quick testing to see what kind of supply it's hooked up to, for example by trying to take 15 Amps and monitoring the line voltage. If the voltage sags too much, then it might back off to 12 Amps; after that, it probably just shuts off. The reason is that all of the AC/DC conversion circuitry is designed to operate with maximum efficiency at a certain input power; if the system can't deliver that power, then it's just going to shut down and assume that there's something wrong with the line.
Note, this does not mean that the battery requires 1kW continuous power to stay charged. In plugged mode, the car will switch the charger on and off every few minutes (if it's at all like a laptop or phone) and draw down the battery ever so slightly in between. This is the most efficient way to operate, and is much better for the battery than constantly stuffing it with a trickle of current.
If you want to see this in action on a consumer electronics scale, try plugging your iPad into the USB extension port on an iMac keyboard -- it will kindly let you know that it's not charging, and it will happily sit there and run its battery down to the cut-off point while plugged in.