Thanks everyone, this is all great! The thing that makes me think hardware is that it's so binary; either it works or it doesn't, sort of randomly. Then switching out and back to the network makes it work again. I've sort of disregarded what the online support person said. She said there was an entire menu item missing on the Mobile Data menu, someting like "Roaming profile". I can see a setting being wrong, but a whole menu item gone from a screen? Hard to see. And the instore guy said that wasn't it.
He did reload the network settings from a PC, so I've done one hard factory reset (had to load all the apps again, very much a pain!) and none of that did anything.
This "RF Box Test" may be what the online support person said I should get the store to do, I think she called it a "Flight Test". But they didn't do either. Guess I'll head back to the store (trying to wait for it to mess up again, but since I WANT it to mess up, it's worked flawlessly the last 3 days!).
Thanks everyone. This is very helpful. Much more help than I've gotten on my last few phones after 40+ calls to Sprint (my older HTC EVO 4G was MUCH worse on Sprint, but that's a different story!).