Auto-delete backups?

Hey all,

I have been having issues recently where my server will hit 100% disk space and sites stop working and what not. 

it will not delete the previous backups. I then have to ssh into it and delete the backups. Is this a new issue or is there a setting I can change to delete these? Alternatively, is there a script/cron job that any one has to delete all except for today and yesterday's backups? 

I copy them off the server to AWS S3 daily although the time doesn't always line up so I'd like to keep the current day+1 just to be safe.