I am running BackupPC, which backs up my laptop via rsync to another computer daily. Suppose that I upgrade one or more packages during the backup process. Is it possible that the package will be broken in the backup because some of its files came from the old version and some from the newly-upgraded version? How likely is this?
Running a backup during an upgrade is not recommended. A typical case when you need your backup is the case when an upgrade is broken. Therefore I recommend to backup after you checked your upgrade is OK.
However this depends entirely on which files are backuped. If it's /home only then running a package upgrade at the same time may probably not do much harm.