First, are you using POP email accounts that you download email from to your computer? If so, login to your cpanel, click the Mail icon and then select Manage/Add/Remove Accounts. You will now see a list of your email accounts. Underneath the the list of accounts, you'll see the "Show Disk Space Used" link. Click this link. You'll now see how many megabytes of email are stored in your account on the server and what the limit is on your account. The default limit is 10 MB, but I recommend you increase that. If you are regularly downloading your email to your computer, you can delete the quota. Just make sure you are deleting all emails off your account, or they can take up all your space.
If the amount of space used is the space or within 2 MB of the total quota amount (or total disk space amount) your problem is the result of too much space being taken up in your email account or your hosting account. You may need to login via webmail to delete emails. Or, if you have been downloading them, check to make sure that your email client is configured to remove emails from the server after downloading them.
If your entire hosting account is full, you will need to delete emails or files from the server. If you cannot access the files to delete them because the account is full, submit a support ticket requesting a temporary upgrade to a larger plan. You can then use that time (typically less than 24 hours) to delete the unnecessary files and download your emails.
Are you using email forwarders? If this is the case, check the Manage/Add/Remove Accounts section to make sure you do NOT have a POP account setup. If, for example, you have firstname.lastname@example.org setup as a forwarder AND a POP account, some of your email will be forwarded and some will be trapped in the POP account. Download your emails from the POP account (or delete them via webmail) and then remove the POP account.
Are you receiving an error message in your email client (Outlook, Thunderbird, etc.)? If so, then you need to submit a support ticket, including the full error message you are receiving. We can help you determine what the error means and provide you with links to information on how to fix the problem. Unfortunately, we cannot repair your email client for you, if that is what is required.
Now that you have fixed the problem, where did the emails go? Mail is spooled on the server when delivery is not immediately possible. The system will retry for a period of up to five days to deliver any mail that it has in the spooler, and then will return the messages to the sender with a note on the undeliverable status.
Those notes, though, sometimes don't get where they need to go, depending on the filtering on the other end, or within the email client of the person to whom they are being delivered. If you believe that you should have received a message from someone but did not, drop that person a note and ask them to resend.