This downtime was caused by... the software we installed prevent downtime.
Yes, you read that right - the software we installed to prevent downtime actually ended up causing it.
It created huge logfiles which, over time, filled up a crucial partition of our hard disk, causing our other software to fail due to lack of free space in which to save files.

I'm starting to believe in Murphy's Law.

On the bright side, I've completely disabled this problematic software, and altered the way we handle log files, so this should never happen again.
I'm also going to set up a new system to detect, and help resolve, any future site issues.