If you get a message informing you that your account has hit a disk limit or that the server has run out of space , this means that you have gone over the limit of storage space that has been allotted to your account. Obviously at this point you are not receiving or sending out messages.
Attachments are the biggest space hogs on the system.
Save any attachment to a location that you have access to, then delete that message. This task alone could alleviate the problem. Look at your oldest messages. Are these messages you really need to keep. If you need to keep messages for the long term, try saving them as text files to a location you have access to.