Does "just-barely-out-of-the-80's" count?
The worst nightmare I ever ran into regarding backups was one which I had not been responsible for doing (which covers 98% of backup problems, I imagine, as to which you good fellows can attest).
I got dispatched to the home office of a restaurant chain here in the Southeast with complaints of "It won't work and we tried everything." I was suspicious that I was walking into a huge mess just from talking to the lady for 2 minutes on the phone but my boss had every confidence that I could dispatch with their problem in no time at all.
So I arrive. Their "server" was an aged 486 (which, admittedly, was perfectly capable of being a server back then) stuck under the lady's desk upstairs and their "backups" were done "automatically" by the system via writing to a DVD drive. All the quote marks indicate things that were not that they were named, naturally.
The problems: They told me it ran Windows something-or-another but it was Unix, it was crashed so hard the logs were not readable (HDD had only tipped 96% over, not dead as in really dead yet) so I had no idea where to even start with a fix, and thinking maybe this backup, which was run "every day" she told me, could help me with at least a start, after replacing the drive
They were re-writeable, there was 7 of them, rotated daily. I pulled the DVDs and put them one at a time in my laptop's drive and each was _completely_ blank. So I had to be the bearer of that most horrible news "Your data is not only gone, it's gone for good." Some other local tech vendor had charged them a princely sum for this useless "backup system".
Bright side, the owner, who had shown up during all this placed the blame 100% on them and not me (I was shocked at that part) and even though they didn't use us for their IT needs later, we did handle her and her husband's systems at home which paid better than that nightmare at the home office ever would have.