They do. They also have a low voltage cut out which might be an issue but should show an error on the LED if it is installed and working properly.These things likely draw more current as the voltage drops. Yeah, check all the connectons. "It's always the connections."
I'm thinking that more specifics about how the system is behaving might make speculations on the problem a bit less random.
FWIW, the thermostat is easy to test. I don't recall the voltages on the circuit so be careful. The thermostat can be shunted out at the switch inside the box or at the controller. Short the connections, leave the box shut for a while and see if it keeps getting cold. If the thermostat is working but can't be set cold enough it can be adjusted (there's a set screw) or, as a short term hack, by moving the sensor to a warmer part of the plate.
Okay, googled it. Here's some official looking stuff.
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