DIAGNOSIS The Nest is experiencing a weak or unstable Wi-Fi signal, often due to distance from the router or interference from the metal HVAC plate.
FIX Go to Settings > Technical Info > Network. Check the signal strength. It must be 'Good' or 'Great'. If 'Poor', move your router closer or install a Wi-Fi extender. Ensure your router's 2.4 GHz band is enabled, as Nest does not connect to 5 GHz networks.
IF THAT DIDN'T WORK A software bug can cause persistent drops. Perform a full restart: Settings > Reset > Restart. If problem returns, perform a full factory reset: Settings > Reset > All Settings. You will need to set up the thermostat again.
HEADS UP During setup, the Nest may connect to a different SSID if you have a mesh network with the same name for 2.4 GHz and 5 GHz bands. Temporarily split the SSIDs during setup to force connection to the 2.4 GHz network.