So you are doing automated backups every day ... in the morning? I hope wee hours of the morning and not prime time for usage of other courses/system. Failed only on Monday and on Friday.
We've not mention specific system details ... maybe it's time ... shared system? How many other host on it? Is the shared system a cheapo package where the provider has strict limits on processing power, space, data throughput, etc. on each customer? True dedicated server or a VPS? On a VM?
but if that is a pattern ... failure on Monday and Friday ... that sorta points to something happening at that time that sucks up all resources and causes the failure to occur ... a restart of something which kills the cron job backup process? Database on same server or is it a remote DB server?
So what else do you have scheduled in Cron?
Is this server in a Virtualized environment?
And there are no hints in error logs of server ... apache error log, php error log, secure log, whatever daemon running log?
Had a server in a Virtual environment once that for 5 minutes of every hour at exactly 32-37 after the hour would become un-responsive ... server up, apache running, mysql running, but no one could access. Finally got the virtual server administrator to look into it. Problem solved ... nothing wrong with the guest OS upon which Moodle server was running ... it was some other guest OS on the same physical Virt server that had been granted use of all resources every hour at 32-37 after the hour to do whatever it was doing.
There's just too many folks that do run autobackups and don't have issues (yet). At least none like your description, so without live monitoring of the system at around the times the autobackups are to be run, just about anything we talk about/suggest is a guess. :| Lot's of what appear to be un-related questions and maybe side trips (inspection of the remaining temp directories, for example).
'spirit of sharing', Ken