The screen goes black and renders the device useless on affected phones. Motorola has identified a fix, and is shipping working devices out and repairing the problem phones. If you’re seeing this problem, best to contact AT&T or Motorola directly so they can send you a working device.
The carrier has changed the Nexus 6 more than any other carrier, baking in AT&T specific “features." The AT&T version of the Nexus 6 includes a locked AT&T SIM card, and most annoyingly won’t let you tether without a valid plan. Another minor gripe is that it has AT&T ringtones included, although the stock Nexus ones are there also. The company logo is also on the back. Many users buy a Nexus to get a pure, unbranded Android experience, which AT&T is arguably not delivering.
The technical problems may or may not have to do with AT&T's modifications, but it seems like an awfully big coincidence that other carriers, who haven't changed the Nexus 6, aren't experiencing the same issue.
Why this matters: If you want an unlocked phone free of any carrier interference, then buy the Nexus 6 directly from Google or Motorola (when they come back in stock). Now that Google’s flagship is on all the major carriers, it’s not unlikely to see them take certain liberties with how it works. While Android purists may be annoyed that Google’s original vision of an untarnished Nexus has morphed, it reflects the realities of working with the major U.S. carriers.