It appears that there are a few issues being discussed here, but a number of people described enough details to make me think a few of us are having the exact same issue.
This is my guess as to the significant factors that identify the issue my wife and I had:
1) Data works fine for about a day. We don't have LTE here yet, but 3G was reliable.
2) After connecting to wi-fi at home and turning it off, 3G/4G data never reconnected.
3) Battery removal, reset, settings manipulation doesn't fix it.
4) After turning on the "Mobile Network" option on the "Wireless & networks" settings page, the text under that option briefly shows "Turning on..." and then either hangs there (greyed out), shows "Use phone for data connection when Wi-Fi is unavailable", or flickers between the two.
5) A hard reset fixes the issue.
Another person posted ealier that he and his wife had the same symptoms, and a few others, so this obviously isn't a fluke or defective device.
Doing a reset after only one day isn't a disaster, but very annoying since my wife and I both spent a lot of time that first day customizing it. It'll be a few months before I stop worrying that it might happen again.