Spent forty minutes on the phone with Apple earlier, because I got a pop-up thing that said my Airport/Time Machine had its name changed or some such thing and that it had no back-ups at all and would have to do a new one from scratch and to make sure no one had somehow gotten into the system and was trying to glom onto it. So, that was a worry and I didn't do the back-up. I ultimately got a Time Machine expert on the phone and she was great - we did a Malware Bytes scan of the computer and it told us everything was safe and sound, so that was good. She had me check a lot of other things, too, just for safety's sake. She ascertained that the device is ten years old and said it could have just been a glitch in the mechanics. On my computer, everything looked right - it was named properly, there was only one device listed, etc. So, she felt safe in doing the new back-up, which it's been doing ever since. I suspect it won't finish that until either late tonight or morning, as that's over 500 gigs o' stuff to do. I'm going to buy a new 3 terabyte back-up drive and back-up the entire computer to that, too, just to be safe.
