What can be wrong when automated backups stop being sent to email?

If you’re using the WP-DB Manager plugin, go into its options and create a backup manually. If this fails, inspect the error message and try to figure out what might be going wrong. If you can’t figure it out, post the error message here.

If the manual backup goes fine, there’s probably an error with the e-mail setup. Check all the e-mail settings and ensure that they are correct. What e-mail provider are you using to send the e-mails (what SMTP server)? What e-mail provider are you using to receive the e-mails (what POP3 og IMAP server)?

If you’re not using WP-DB Manager, we need to know how you’re doing these backups and all the information you’ve been asked to provide by me and the others.