cannot restore backup to iphone

I've had an iphone 4 since they first came out and finally decided I would upgrade it to IOS7. I did a backup through itunes, installed the new IOS and then tried to restore my backup to the device. Itunes went through the motions then at the very end it said "...could not restore ___ because of an error". It didn't give a number for the error message. I did a lot of internet searching, tried it many times, even loaded itunes on another computer and tried that too, with no luck.

I figured I had had the phone for a long time and deserved a new one so today I went to the ATT world headquarters and got a new iphone 5s. I plugged it into my computer and tried to restore the data. Itunes went through the motions, and at the end gave me the same error message. So I've gotten the same error message many times, with 2 different IOS devices on 2 different computers. I don't understand what the problem is or what I can do to get my data restored.

You very likely have a corrupt backup stored in iTunes. No telling how it happened? Did you also backup to iCloud? If you have no iCloud backup and the iTunes backup is corrupt, there's nothing you can do but re-download your apps and music from the iTunes store. You will not have to pay for them again. If you had apps from other than the iTunes store, you may be out of luck.

Actually, iTunes probably has multiple backups of your iPhone 4, so maybe restore from an older backup than the most current one and see if that fixes the issue.