I was having a similar problem.
With router set to N only, Nexus refused to connect to internet.
With router set to B/G/N mixed, Nexus connects fine but other wireless devices in the house are unable to connect.
With router set to G only, all devices work great... although I'm stuck with using Wireless G speeds.
However, I don't really mind the G speeds as it's still plenty fast for anything I do on a tablet or my other wireless devices. Whenever any heavy downloading is done... it's done on a wired PC... so no big deal.
I don't know if it's an issue with the wireless router I have or what... but eventually down the road I will upgrade to a quality wireless router. Right now i'm using this POS router/modem combo my Internet service provider gave me.
So conclusion: Try setting your wireless router to G only, N only etc... and see which one works for you and allows ALL your devices to connect without problems.
EDIT: Just switched over to N only and now it's working for some reason. But yeah... BGN mixed only allows the Nexus to connect and nothing else. So try one of the "ONLY" modes on your router.