It only happened while it was unplugged (until today, when I hoped that plugging it in would fix it).
Prior to today, I'd only used the wall plug to charge it, except for the very first day I got it when I was activating it. Today and then were the only times I had it plugged into my Mac.
And it was plugged into my keyboard USB, not directly into the back. I have an older iMac (one or two generations from the current ones).
Oh wait, I take that back. I did charge it twice while I was at work on my ThinkPad laptop.. but i did the same with my 4S and there were never any problems with that one.
By the time I got to the Genius Bar, not even a wipe and restore from factory settings would resolve it, so I got a new one. They didn't seem to have any idea what may have caused it. But it was definitely pretty much a brick, albeit a brick that still powered on. And still got good LTE/4G signal.
Let's hope this one lasts longer.
Question for those who are having issues - what color and what GB size are yours?
Mine is white 64GB.
Just wondering if there is any relation to color/size (doubt it, but worth asking).