It's probably because the drive has run out of room. Remember that Time Machine keeps multiple backups, so Kathy may just need to delete an old backup or get a larger hard drive.
She's also getting an error that says the drive is read only. Leo says that he's gotten that message as well. If she did a hard shut down, that may cause the read only error to occur. She should try and repair permissions first. It's possible that Kathy will have to start over and rebuild her backup from scratch, though.
Leo also advises using Super Duper to do her backups. The benefit there is that it won't be vulnerable to hard shut downs, and it also will allow her to make her backup bootable.
The Chatroom says that OS X will do a File System Check (FSCK) and if the cached permissions were wiped out, then the disc gets mounted as read only.