The problem was probably in the wiring. We had a Nest that was not properly wired. We switched over to a new thermostat that was designed for Nest and it worked great. We moved the old thermostat over to the new Nest, and it worked great too. We have since replaced the old thermostat with a new one that has the same wiring, and the Nest still works great. It is just a matter of time before we find out why these issues started.
Although this might seem a bit extreme, I would not be surprised if this problem is a result of some bad wiring or other issue. The thermostat is always on at the same time, which is usually the same time the AC is. This is a known issue that can be fixed via a service call, but we’re not too worried about it. As long as the AC is on, it should be fine.