Backup failing at preparing stage every time

My Mom got a new (refurb) aluminum MacBook 13.3" and a new (not refurb) Time Capsule (Dual). The Time Capsule I believe is fine because it works for the internet, we are able to mount it as a drive, and I was able to start a backup to it from my older non-aluminum MacBook, but every time we try to backup from my Mom's aluminum MacBook it fails with the error, "Time Machine Error", "The backup disk image could not be created."

Here is the log from a run (I've x'd out her last name):

Apr 25 19:48:15 Macintosh /System/Library/CoreServices/backupd[287]: Backup requested by user
Apr 25 19:48:15 Macintosh /System/Library/CoreServices/backupd[287]: Starting standard backup
Apr 25 19:48:16 Macintosh /System/Library/CoreServices/backupd[287]: Mounted network destination using URL: afp://Grace%20xxxxx@Grace-xxxxxs-Time-Capsule.local/Data
Apr 25 19:48:16 Macintosh /System/Library/CoreServices/backupd[287]: Backup destination mounted at path: /Volumes/Data
Apr 25 19:48:16 Macintosh /System/Library/CoreServices/backupd[287]: Creating disk image /Volumes/Data
Apr 25 19:48:22 Macintosh /System/Library/CoreServices/backupd[287]: Error moving disk image from /Volumes/Data.tmp. to /Volumes/Data - Error Domain=NSCocoaErrorDomain Code=512 UserInfo=0x5257b0 "“Data.tmp..sparsebundle” could not be moved to “Volumes”."
Apr 25 19:48:22 Macintosh /System/Library/CoreServices/backupd[287]: Failed to create disk image
Apr 25 19:48:27 Macintosh /System/Library/CoreServices/backupd[287]: Backup failed with error: 20
Apr 25 19:48:28 Macintosh /System/Library/CoreServices/backupd[287]: Ejected Time Machine network volume.

I note that in the /Volumes directory it leaves behind a directory called "Data.tmp..sparsebundle"