The amount of mailbox storage included depends on the the plan that you choose. This problem usually occurs if you do not delete messages from the server from time to time.
If you are using a standard email client, like Outlook, messages are normally removed from the server and stored on the PC when you check mail. However, there is a setting to leave mail on the server, which comes in handy if you check mail on more than one PC. If you do this, be sure the email program on one of the PCs is set to remove mail from the server or you will exceed your quota over time.
If you are using Webmail as your only way to view email, messages will remain on the server until you delete them. To conserve storage space in your email account, make sure you empty your trash folder on a regular basis.