In any/all postings of issues, would be nice to get some basic information concerning system: OS, standalone/remotely hosted, etc.Does make a difference in answers. So given the info you've shared:
In a 1.9.19+ version of Moodle, there are a few tables related to backup:
mdl_backup_config - this table is related to the backup settings - ie what to include in the backups.
mdl_backup_courses - course id, laststarttime, lastendtime, laststatus, nextstarttime (time fields are in UNIX epoch time)
mdl_backup_log - contains course ID, time, laststarttime, info - when one runs manual backup, same information is displayed in the browser window.
Backups have to be built. Moodle uses /moodledata/temp/backup/ as the build area. With whatever you have to browse files on the system, check that folder out. There might be .log files which could provide clues. Also, if running automated backups to an alternative directory, check that directory for backup files. Moodle uses 'copy' rather than 'move' routine to take the built backup from the building folder to the destination folder. With some operating systems, the copy routine cannot copy a zip file larger than 4 Gig. The backup files are there and can be moved (manually) to another location.
IF your moodle is on a system that has partitions, is the partition upon which the alternative backup location full?
'spirit of sharing', Ken