When your mailbox is getting full, you and your administrator will receive an email notification specifying your mailbox utilization percentage.

By then, you may choose to clear your emails by download-and-delete through your mail client, or login with your webmail access to delete your emails.

Note that when your mailbox quota is full, there will be a period of penalty time even after you have cleared off the emails.
The penalty time is required so that the email server can work more efficiently.

If you are urgently in need of receiving any incoming emails, you may consult your administrator to increase your mailbox space.

Do ensure sufficient quota has been allocated for the user's account, especially for the more active ones, the ones that are getting constant notification on mailbox getting full.