The thermostat responds to engine temp, not radiator temp. If the cooling system is not able to cool the water fast enough to cool down the engine, you will either have a hot running engine, or it'll over heat completely and blow all the coolant.
The job of a thermostat is to keep an engine at a pre-determined temperature. When the engine is cold, the thermostat is closed, preventing the water from circulating. When the engine temp reaches the same temp as the thermostats rating, the thermostat opens to begin cooling the engine down.
Normally, the thermostat will open a small amount, allowing cooler water from the radiator to enter the hot engine and cool it down. Once the engine cools down some, the temperature sensitive spring will close the valve in the thermostat and wait for the engine temp to rise again to the temp that the car maker requires. Its normally and endless cycle of "engine heats up, thermostat opens, engine cools down, thermostat closes, engine heats up" and so on.
In an ideal world, the cooling system will be adequate enough to keep the engine from over heating.
Once the engine temp and the radiator water temperature exceeds the rated temp of the thermostat, the thermostat will stay completely open until the engine is finally allowed to cool down, or it will over heat. At this point, about all you can do is shut down the engine and wait for things to cool off.