Yes a bad thermostat on a vehicles engine can cause the coolant to leak from other areas due to back pressure in the system if the thermostat is stuck in the closed position.
The thermostat opens up as the coolant gets hotter to allow the coolant to flow through the engine properly to keep the engine cooled down and when it does not open up it can cause the coolant to leak out from other hoses or can cause the coolant to boil and possibly come out of the coolant tank.
If your engine is overheating or you're noticing coolant leaks when the engine gets heated up then it can potentially be the thermostat not opening up.
Also check the radiator cap and check the hoses and water pump for any leaks or wear on the radiator hoses, heater hoses etc.