When a mailbox reaches 100% of its space, it stops receiving messages and may stop sending them. Check the mailbox usage in cPanel and, if it is at the limit, increase the space from Email Accounts.
Also check the plan's total space in cPanel, under Disk Usage: if the hosting account has run out of space, email will stop coming in and going out even if each mailbox still has available quota.
If you use POP3 and your client does not delete messages from the server, webmail will fill up over time. In that case, we recommend using IMAP, which keeps messages synchronized on the server. The differences and ports for each protocol are listed in POP3, IMAP, and SMTP: email differences and ports.