I have had a problem on my Mac mini for several months. Time Machine hangs after backing up a small amount of data, and refuses to continue. I can leave the machine running for days and nothing happens. This is the sort of thing I see when I click on the Time Machine icon in the menu bar:

Time Machine stalled - the figures in the screen shot never change
It is rather strange as we have an iMac running in the same network, on the same version of OS X (Snow Leopard – Version 10.6.7) and Time Machine runs just fine on that Mac.
By chance I have found the solution to the problem, and I haven’t seen a description of it anywhere on the internet, so I thought I’d post it here.
Our configuration is that we have a (wired) network with the two Macs and a Synology DS210+ NAS server connected. The DS210+ stores all our data, iTunes music and so on and when the Macs boot, several disk volumes on the DS210+ are automatically mounted on their desktop. These volumes are all stored on the internal drives of the NAS. We also have an external drive attached to the NAS via USB, which is exclusively used by Time Machine for its backups.
Solution:
If, on the Mac mini, I unmount the volumes stored on the internal drives of the NAS, Time Machine immediately continues and completes the backup. If I remount any of those volumes, Time Machine stalls indefinitely until I unmount them. Simple. It works every time. I have no idea why I only need to do this on the Mac mini, and not on the other machine.
If you find this useful, I’d appreciate a short message either as an answer to this post, or by e-mail (see the “Contact us” page) – I’d be interested to see if others find this solution works for them too!