Email and Spam Assassin are the two most likely culprits
If you are running Spam Assassin, you MUST remember to login to Horde Webmail occasionally to clear out your spam box, OR log into cPanel and go to Mail -> Spam Assassin, and click "Clear Spam Box". Otherwise it will simply keep accumulating spam until it eats all available disk space.

Once this happens, here is an easy way to clear out a spam box fast without having to login to Horde and delete each spam individually:

Go into your cPanel and click on the File Manager icon.
Click on the folder icon next to the word "mail" - DO NOT CLICK on the word "mail"
If you see a file called "spam", click on the file name. Then at the upper right on your screen, click on "Delete File".
If you see other folders (such as one with your domain name), open the folder/s by clicking on the folder icon/s and and look inside for a file called "spam" in each one of these too. If you see any files called "spam", delete each file you see called "spam". DO NOT DELETE THE ACTUAL FOLDERS. IF YOU CLICK "DELETE" WHEN YOU ARE ON A FOLDER NAME, YOU WILL DELETE THE ENTIRE FOLDER. Make sure the upper right corner of the screen shows "spam" for a file name before you hit "Delete File".

If it is your inbox (not spam box) that is completely stuffed and eating up all your disk space, you can do one of the following:

Check your email program settings where you receive your domain email, and be sure to uncheck the box that says "Leave a copy of messages on the server". (To do this in Outlook Express you would go to Tools -> Accounts -> Mail -> Properties -> Advanced.)
Then check your mail with your email program and allow it all to be downloaded onto your computer. Warning: This can tie up your computer for a long time if you have a lot of email in your inbox on the server. As long as you are NOT keeping copies on the server, your mail comes off the server as soon as you download it onto your computer. After clearing mail off the server, if you want to keep future incoming mail on the server again, just go back to your email program and re-check the box to leave messages on the server. However specify a shorter time to leave mail on the server next time.

If you don't care about keeping or reading ANY of the email stuffed in your Inbox/es on the server, follow the same procedure as outlined above for spam boxes, except delete the file/s called "inbox". This will PERMANENTLY delete all currenly-existing mail in those inboxes. Be careful not to delete the folders themselves. Make sure the upper right corner of the screen shows the inbox file name before you hit "Delete File". The inbox files will be automatically recreated for you the next time you receive email.

It could also be possible that the quota set for the email address has been reached
The e-mail account must have disk space available in order to receive e-mail. In order to receive mail you have to set the quota to anything above your current limit or 0 which is unlimited or rather limited by the space available for the hosting account.

Another possible culprit is using "Full Backup" in cPanel Backups
If you want to make backup copies of your site, it is strongly recommended you use "Home Directory Backup", NOT "Full Backup". With Home Directory backup you can save a copy of your web site as a zipped file onto your own computer. Once you have a Home Directory backup, you can use it in future to restore your entire web site by browsing to your file from cPanel and then clicking "Restore".

If you use cPanel's Full Backup function, it just puts a zipped file of your web site into your root (above public_html) and is NOT accessible from cPanel for restoring your site. The only way to restore your site from the Full Backup is to submit a support ticket and ask us to do it. Meanwhile the backup sits in your account, taking up a lot of space. You can see any backups you have made this way by going into cPanel -> File Manager and looking for zipped files near the bottom of your root directory list of folders & files.

If it is Full Backup/s that are taking up space in your account, we suggest you delete these using cPanel File Manager and then immediately go to cPanel -> Backup and make Home Directory backups (and if applicable, separate MySQL backups and Filter/Alias Backups). You can download each of these to your own computer instead.