Time Machine won't back up onto Time Capsule

Question

I upgraded to Mountain Lion last week, and Time Machine won't back up to my Time Capsule. When it fails, I get the message:
Time Machine couldn't complete the backup to "Tom's Time Capsule".
The backup disk image "/Volumes/Data/TomsMac.sparsebundle" could not be created (error 60)
I sometimes get the same message, but with "error 17"
I also get the following error message popup (even though the disk is the one in my Time Capsule):
The disk you inserted was not readable by this computer.
The popup offers three choices: Initialize, Ignore, Eject.
My local Apple Store Genius concluded that the problem must be with the Time Capsule, so I bought a new one. That didn't fix the problem.
Time Machine Buddy messages:
Starting manual backup
Attempting to mount network destination URL: afp://Tom%[email protected]/Data
Mounted network destination at mount point: /Volumes/Data using URL: afp://Tom%[email protected]/Data
Creating disk image /Volumes/Data/TomsMac.sparsebundle
Error 60 creating backup disk image
Failed to create disk image /Volumes/Data/TomsMac.sparsebundle, status: 60
Backup failed with error: 20
Failed to eject volume /Volumes/Data (FSVolumeRefNum: -108; status: -47; dissenting pid: 0)
I did Repair Disk Permissions and Verify Disk and everything's OK.
I erased the Time Capsule using AirPort Utility and did a hard reset on the Time Capsule.
I did a re-install of Mountain Lion and that didn't fix anything.
My MacBook Pro is accessing the Time Capsule wirelessly, as it always has.
Help!

Answer 1

Just to be sure, we're talking about a Time Capsule -- not an AirPort Extreme connected to an external hard drive. Right?
If so, proceed as follows.
Please read this whole message before doing anything.
This procedure is a diagnostic test. It’s unlikely to solve your problem. Don’t be disappointed when you find that nothing has changed after you complete it.
The purpose of this exercise is to determine whether the problem is caused by third-party system modifications that load automatically at startup or login. Disconnect all wired peripherals except those needed for the test, and remove all aftermarket expansion cards. Boot in safe mode* and log in to the account with the problem. The instructions provided by Apple are as follows:
Be sure your Mac is shut down.
Press the power button.
Immediately after you hear the startup tone, hold the Shift key. The Shift key should be held as soon as possible after the startup tone, but not before the tone.
Release the Shift key when you see the gray Apple icon and the progress indicator (looks like a spinning gear).
*Note: If FileVault is enabled under OS X 10.7 or later, or if a firmware password is set, you can’t boot in safe mode.
Safe mode is much slower to boot and run than normal, and some things won’t work at all, including wireless networking on certain Macs.
The login screen appears even if you usually log in automatically. You must know your login password in order to log in. If you’ve forgotten the password, you will need to reset it before you begin.
Test while in safe mode. Same problem(s)?
After testing, reboot as usual (i.e., not in safe mode) and verify that you still have the problem. Post the results of the test.