I recently discovered that my email address mailbox had reached its limit, which is why I wasn't receiving email addresses (for over a week). I've now changed the quota to unlimited, but I haven't received the backlog of emails yet (and the senders never received a bounce-back) so I was wondering if I will receive them and if it's possible to have them pushed from the GoDaddy side?
If your email account was over it's limit and the emails were rejected, those emails wouldn't have been saved or stored. They will not be delivered to your email address. The senders would have to resend the emails.