I had worked at a company ages ago and we learned this the hard way. All backup ran perfectly with not a single error, or so we thought. When a certain backup was needed it was found there was a problem with the head on the tape drive leave all the backup unusable! It's things like this you live through once, learn the hard way and then make sure backups get tested!
I get the feeling that there is more to this story than they are letting on especially because the impact date range ("not updated since Dec 3rd") does not have a "start" date.
Most failures result in the loss of new files, not old files -- notably those created since the last incremental. Did they get really lucky with their backup timing, or is there some sort of hierarchical storage in play?
Also, since they were able to recover 10 of 14 affected groups, it seem more like a bad tape than a bad tape drive, in which case I would think the "November" backup would likely not be bad in the same spot and could reduce the loss window to "updates made between Nov 3 and Dec 3".