Thanks for the tips.
I had, in fact, looked at those links. That's why I stated the same power adapter, etc and it's how I ended up here. I checked the troubleshooting wiki FAQ and it had nothing on this topic.
I checked my hardware and I have 2 rpi Made in the UK with Samsung 304 chip and 1 rpi Made in China with Samsung 246 chip.
However, it's one of the UK boards that has the issue. Same chip, so it doesn't look like that. (I tried again last night).
As for OS release, I meant that it was up-to-date with the line about "rpi-update done, etc" (though I realize I didn't specifically mention OS apt-get update && apt-get dist-upgrade. - I also did an apt-get upgrade and apt-get autoremove, btw).
So, the only thing I can think of, OS-wise, is that I might need to try a new, fresh debian, rather than the dist-upgrade. Though I'd think the upgrade would work.