Our mail servers operate a system of quotas.
This means that for every user (mailbox or gateway), according to their contracted service level, we set a maximum amount of storage that they are allowed to use on the server for emails.
The effect of this is that when the maximum is reached our servers will no longer accept messages for the user. Obviously we do not want this to happen so we have set the mail server to warn you when you are nearing your quota so that you may either clear down the mail box or purchase more space from us.
This often happens most when a client is either using IMAP for their email, which stores all the messages on the server, or, if using POP, they have set it to leave a copy of messages on the server.