Access your cpanel. In the Mail section, click on Email Accounts.
Scroll down the page to find your mailbox. You'll probably see a solid red bar indicating that the Usage equals or exceeds the Quota.
For immediate relief, simply increase the mailbox quota to a reasonable size. Do NOT set the quota to unlimited as this can result in an extremely large mailbox and possible corruption down the road. About 175 MB should be more than enough for regular use. Click on the Change Quota button on the right.
1) If you are downloading messages to your desktop with POP3 protocol, then make sure your email client is deleting messages on the server at the time they are downloaded. This helps to assure that your mailbox always has enough space for new messages.
When you move messages to the Trash folder using webmail, you then need to empty the Trash folder.
-- To empty your Trash folder in Horde, select Purge Deleted from the Other menu in the upper right corner.
-- To empty your Trash folder in RoundCube, select the Trash folder, then click the "gear" icon in the lower left and select Empty.
-- To empty your Trash folder in SquirrelMail, look in your Trash folder, select all the message and click on the Delete button at the top.
2) Both IMAP and webmail store everything you send and receive on the server. If you are using IMAP protocol or webmail, then regular, periodic maintenance is required to delete old messages in both the IN and the SENT folders (and any other custom folders that you have created) to make sure your mailbox has enough space to receive new messages.
Of course, the best use of web-based email is during brief periods of time when you are away from your regular computer. For ongoing, day-to-day use, a standard POP3 client is best, such as Thunderbird, Outlook, Windows Mail, Apple Mail, etc.