Should backups fail, and they do, there are several places to begin the troubleshooting:
Check the device and the media physically.
Check the device manager and driver.
Check the event logs.
Check the backup log.
If using tape drives, check the media pools.
Check the disk space available for storage both on the media and on the server.
Check the Usenet archives on Google groups.
Check the Microsoft Knowledge Base if using NTBackup or the vendor site if using third-party backup software.
Don't forget simple issues like unformatted media, write locks on tapes, and so on.
Check for conflicting running processes and open files.
Do a manual backup and watch for error pop-ups.
This is not an exhaustive list but hopefully provides some ideas for where to start. Like all debugging, it is half experience, half luck, and half black art.