Have you tried doing a reset? Maybe you have an app that is creating the problem you are experiencing.
Sent from my Galaxy Nexus using Tapatalk
I haven't explicitly done a factory reset, but I did unlock the bootloader, and the problem happened before and after the device wipe. I also relocked the bootloader (in anticipation of exchanging the device) with the same issue.
It's hard to test though, because it doesn't seem to consistently occur

As an aside, the phone doesn't receive any calls in this state either, so it's like the radios are not responding at all.