Since upgrading to 10.6.1 (and possibly before) Snow Leopard has consistently failed to perform Time Machine backups to the Time Capsule on my network. The backup starts but quickly fails with the message “The backup was not performed because an error occurred while copying files to the backup disk.”
After rebooting both my laptop and Time Capsule, and even trying to create a fresh time machine image I finally discovered a way to fix this issue – downgrading to 7.4.1 of the Airport Extreme / Time Capsule firmware.
To do this open Airport Utility, hold down the “option” key on the keyboard and choose “BaseStation->Upload Firmware…” from the menu bar. You should now be able to downgrade your Time Capsule / Airport Extreme to previous firmware versions, and 7.4.1 works flawlessly with Time Machine / Snow Leopard 10.6.1 for me.
Hopefully this will save others from lots of head scratching 
Since upgrading to Snow Leopard, my iMac, (2007), doesn’t perform incremental backups. After the first backup, each subsequent backup is almost the same size, ( 330 GB ), and it does this every hour until I run out of space on the third hour.
My MacBook Pro, (2009), had no such issue after the upgrade.
I use firewire800 connection on both boxes.
Switching to Carbon Copy in the meantime
I had same problem, although with a WD MyBook HDD rather than a Time Capsule, and eventually gave up and bought SuperDuper. Wish I’d found it sooner!
Having the same problem. Downgraded the firmware to no avail. Have you heard of any other fixes?
Here’s the fix that worked for me:
See:
Fix on ready nas
I tried the reverse firmware and no change. But the Spotlight privacy fix was the one that worked.
Having the same issue after upgrading to SL. Your suggestion did nothing for me personally.
OS 10.6.2 (Intel)
TC using FW 7.4.1 or 7.4.2