I hate it when I do several things and I solve a problem, but do not know which "thing" solved the problem. I have not seen a solution on any thread, but I was able to solve my problem so I wanted to share.
**In the original post I forgot to mention doing a disk cleanup, particularly temp files. Windows has a history of having problems when there are temp files; probably opens an existing one instead of creating new.
You may or may not need to make sure that the Volume Shadow Service is running, but I did (a suggestion from the MS Technet forum). By itself, this did not fix the problem:
Start, right click Computer, Manage; select Service. Services and Applications, Volume Shadow Service, and set the service to "Automatic"
Trying to defrag my hard drive I got an error indicating that I had a problem - I'm assuming that this was not a coincidence. Not sure if this is part of the solution (the part I hate), but I could not get Vista to run chkdsk on the boot disk. This was either a quirk or part of the problem - I've done it before. I started Vista and before it completely started I held the power button down for 5 seconds and it shuts down immediately. Keep in mind you are trying to back up your system so this is a really bad time to intentionally crash the system. The smart way is to create a boot CD/DVD and put chkdsk on the CD/DVD; boot from the CD/DVD and then run chkdsk on your boot drive.
When Vista restarted it indicated that it had failed to start correctly and recommended that I allow Vista to scan the disk. Once again, keep in mind that I was asking the O/S that refused to do a backup to repair itself so that I could do a backup.
The good news is that it worked. The repair found and fixed problems (didn't share with me what it repaired unfortunately) and now my backup is working. I am able to back up to both DVDs and an external hard drive.
Experience says that it was running chkdsk to repair a disk problem that eventually solved the problem, but with MS products, one just never knows if it's the process or one step. Could have been the combination of setting the Volume Shadow Copy to automatic and running chkdsk.
Good luck! gwp1948