When you initially created your email account in cPanel you had the option to set the email quota (the size of your email account). The default setting when creating an email account in cPanel is 256MB.
If someone tries to send you an email and your email account has reached its quota they will receive a bounce-back message saying ‘ Mailbox quota exceeded’.
How to fix mailbox quota exceeded error
Login to your cPanel and click the Email Accounts icon.
Click the Manage button beside the email address.
Under the Allocated Storage Space setting you can increase the value shown in the box or set it to ‘unlimited’.
Scroll down and save your changes by clicking Update Email Settings.