DIAGNOSIS Weak or unstable Wi-Fi signal at the thermostat location. The 3rd Gen Nest has a weaker radio than some routers.
FIX Restart your router and modem. Then, on the Nest, go to Settings > Reset > Network. Choose 'Restart' first. If that doesn't hold, do a full 'Forget Network' and re-add it. Ensure you're on a 2.4GHz network band, not 5GHz.
IF THAT DIDN'T WORK The distance or obstructions (walls, metal ducts) are too great. Use a Wi-Fi extender. Place it halfway between your router and the Nest. Set the extender to broadcast the exact same network name (SSID) and password as your main network for easiest reconnection.
HEADS UP If your router uses 'band steering' or a single name for both 2.4GHz and 5GHz, it can confuse the Nest. Go into your router settings and create a separate, dedicated 2.4GHz network with a different name just for the Nest.