For some reason my new GNEX will connect to my wi-fi and the indicator will turn blue. After about 20 minutes the indicator turns gray and I have to manually turn wi-fi off and on again for it to start working..
On your GNex, set up a static IP address in settings (touch and hold the SSID of your router in your wifi settings and go to advanced). That should do it, but it sounds like your router is having DHCP issues. Try updating the router firmware.