Sent email failed, but the email id is valid

Most of the time we have seen that the sent email failed. The reason for this issue is mailbox is full.
TO resolve this issue we need to archive huge number of emails in INBOX

SharePoint recycle bin

Issue:
Some time we observed that the list/library was unable to restore from site recyclebin
Resolution: 
Increase the list throttling and restore the list from site recyclebin.