Both my cousin and I have backups on my PC and we both have the iPhone 4. However, she has Sprint and iOS 5.1.1 and I have AT&T and 6.0.1.
Could I use her backup to downgrade my iPhone to iOS 5.1.1?
Unfortunately the backup files only contain personal data, and don't store the iOS version. Apple makes very difficult, if not impossible, in some cases, to downgrade iOS versions.
Here is some additional information from Apple regarding information that iTunes backs up.