You should look at your backup log and see what Oracle files have been skipped. I would guess that your database is starting to soon and that you are not really backing up your database. You need to find out how long it takes to backup your database and allow enough time for that to happen. You should be monitoring you backup log for any errors or skipped files that might indicate a problem. i.e. If your backup is skipping the system datafile everyday then you don't have a reliable backup.

You can do an export everyday. I think of an export as a safety net. I wouldn't want to have to restore from it, but it is nice to know I have it. When you restore from an export you restore to the time the export was taken. If you are in archivelog mode, and you should be, you can in theory restore up to the point of failure.

You don't have a valid backup, until you have a valid test of your backup on another system.