DIAGNOSIS This is usually a weak or unstable Wi-Fi signal at the thermostat's location, or the Nest's internal software getting hung up. The Nest E has a less powerful radio than the higher-end models, so it's more sensitive to distance and interference.
FIX Start by checking the signal strength. 1) On the thermostat, go to Settings > Technical Info > Network. Look at the signal strength. Anything below "Good" is problematic. 2) If it's low, try changing your router's Wi-Fi channel to one less congested (like 1, 6, or 11 on 2.4 GHz). The Nest E only uses 2.4 GHz Wi-Fi. 3) The most reliable fix is to improve the signal. You might need a Wi-Fi extender placed between your router and the thermostat, but make sure it's a quality one that creates a stable network.
IF THAT DIDN'T WORK If the signal is strong but it still drops, it could be a software bug. Perform a full restart: go to Settings > Reset > Restart (not a full factory reset). This clears its temporary memory. If problems persist, you may need to remove the device from your Google Home app and set it up again as new, which reinstalls the software.
HEADS UP Thick walls, metal ductwork, and large appliances (like refrigerators) between your router and thermostat can kill a 2.4 GHz signal. Also, some older modem/router combos from ISPs are notoriously bad at maintaining stable connections with smart devices.