Time Machine confused, backups don't appear

I normally use Time Machine to back up my MacBook on one external drive at my office, and on another at home. But since yesterday it is completely confused and not backing up on either. My home drive does not appear on "Browse Other Backup Disks" any more, even though running Disk Utility on it shows that is fine. Similarly, it stopped backing on on my office drive as well. What the heck is going on? And what can I do about it?