So as soon as one new datafile was added to the database (I estimate, a week after that regime was put in place) all subsequent backups were invalid. We were in that situation for over a year and everyone believed that the databases was being properly backed up. We would even occasionally restore a file (picked from the backupset, hmm) and block-verify it. I only discovered it when I was upgrading the system and I fixed it real quick (dynamically generating n equally-sized lists of files is trivial!). Hard coding the list of files was stupid, sure, and betrayed a fundamental lack of understanding about how the system worked - but it was also stupid that no-one ever checked, and that betrayed a lack of understanding of how organizations work.
The lesson I took away from that was, you have to have people who have visibility of things end-to-end.