Made that mistake once. I was interning at the IT Helpdesk at a Pharma Startup. Our CEO calls us one day, informing us that he deleted a slide deck, and was wondering if we could do anything about it. We dutifully attempt to restore from tape, to find the tapes blank and that our backup has done literally nothing for the past month. Thankfully, the data loss was nothing more severe, but the lesson stuck.
In my last job I was a member of an IT team of three, one guy was the CIO + he did sysadmin and IT support for an office of about 60. I did sysadmin and support for about half a dozen offices, 200 odd staff, in two countries with a geographical spread of several hundred kilometres. The third guy was support and sysadmin for our US office which had about 12 people (cushy job). I popped out there once for a week or so to help train the guy on our new systems. They had an old backup system that copied diffs to disk which he had to take home every day, but he didn't like doing that so they invested in a very expensive off site system. I was having a dig around when I was over there and it appeared to me that the diffs hadn't been happening. I confronted the guy and he said that he hadn't set it up yet and knew that would bite him in the ass eventually. They hadn't run any backups for six months, but he had been giving positive reports to both his manager and the rest of the IT team about the hard work he was putting in setting it up, and how it had been so reliably backing up, we celebrated when the first full copy of data finally finished, etc. Needless to say he lost his job.
We were using an external vendor for our production database (I've changed this now) and I was poking around their management console one morning when I notice something (I was about two months into this gig). The CTO walks into the office at that point and I ask him: "Hey, I can see that the staging database is being backed up, but the prod database doesn't have any backup files. Where are the backups for prod?". His response: "Press that manual backup button immediately, please." Turns out that production had been running for nearly three years without backup actually running...