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