I am still having problems so I have installed the dashboard widget "Time Machine Buddy" to try to get more information. It reported the following:
Starting standard backup
Attempting to mount network destination URL: afp://Rosemary%20Burden@Time-Capsule.local/Time%20Capsule
Mounted network destination at mountpoint: /Volumes/Time Capsule using URL: afp://Rosemary%20Burden@Time-Capsule.local/Time%20Capsule
Failed to eject volume /Volumes/Time Capsule (FSVolumeRefNum: -104; status: -47; dissenting pid: 0)
Waiting 60 seconds and trying again.
Network destination already mounted at: /Volumes/Time Capsule
Failed to eject volume /Volumes/Time Capsule (FSVolumeRefNum: -104; status: -47; dissenting pid: 0)
Waiting 60 seconds and trying again.
Network destination already mounted at: /Volumes/Time Capsule
Failed to eject volume /Volumes/Time Capsule (FSVolumeRefNum: -104; status: -47; dissenting pid: 0)
Giving up after 3 retries.
Backup failed with error: 21
I don't understand this information but I'm a little puzzled to see references to my wife's name in there (this is her computer really and I'm trying to ensure her account is backed up) as this is not the name of this computer (it's called "Silver Macbook Pro" and the sparsebundle on the time capsule is called "Silver Macbook Pro.sparsebundle.)
In the middle of writing this, it actually backed up correcly, and now the report from Time Machine buddy says this:
Starting standard backup
Network destination already mounted at: /Volumes/Time Capsule
QUICKCHECK ONLY; FILESYSTEM CLEAN
Disk image /Volumes/Time Capsule/Silver Macbook Pro.sparsebundle mounted at: /Volumes/Time Machine Backups
Backing up to: /Volumes/Time Machine Backups/Backups.backupdb
1.40 GB required (including padding), 379.94 GB available
Copied 3860 files (4.6 MB) from volume Silver MacBook Pro HD.
1.39 GB required (including padding), 379.94 GB available
Copied 613 files (1 KB) from volume Silver MacBook Pro HD.
Starting post-backup thinning
Deleted /Volumes/Time Machine Backups/Backups.backupdb/Silver Macbook Pro/2011-10-26-215858 (484.0 KB)
Post-back up thinning complete: 1 expired backups removed
Backup completed successfully.
Ejected Time Machine disk image.
Ejected Time Machine network volume.
Now my wife's name does not appear, and it works. What caused the change? I don't know.