You receive this message when a mailbox on the server of your hosting provider has used up its space quota. You can deal with this problem as described here:
Do you keep receiving e-mails with the notification ‘Mailbox full’ although you are forwarding the e-mails to an external account? If you simply want to forward e-mails, you do not need to have a corresponding account (except for the forwarder) on the server. Set up a forwarder only. This frees up additional space that you can use for your website.
Have you not configured a forwarder for this e-mail address and the space in your mailbox is used up? Please check
To guard your account from becoming clogged with spam, you can enable the Spamscanner with its Spambox feature. This will identify undesired advertising e-mails and automatically delete them after a certain period, thereby freeing up this space again.
If the size of the mailbox is still insufficient after cleaning it up, you can always increase your e-mail quota from the Control Panel to create more space for messages. Please click here for more information.