If you're getting "mailbox full" errors, it's probably time to do a little housecleaning. Your email service comes with a few size limits. When you exceed those limits, you cannot send or receive emails until some are deleted.
To solve this problem:
Messages in your Sent, Drafts, Deleted/Trash and Outgoing folders all count against your size limit so be sure to clean out everything you can. Sometimes it's not the quantity of messages but the size. Messages with large attachments, particularly ones containing images and/or video, can rapidly consume your mailbox's storage capacity.
To confirm you've freed up enough space, try sending yourself a test message from a different account.
Did you know? People sending emails to you can also be affected by your full email box. They may receive an "undeliverable" message if there's no room in your box for their message to fit.