Email apps stay connected in the background so new messages arrive on their own. When that connection breaks, the app quietly stops pulling mail and you only notice hours later.
The most common break is authentication. If you changed your email password on the web but not in the app, the app keeps trying the old password and the server refuses every request.
A full mailbox produces the same symptom. Once you hit the storage limit, the server stops accepting or delivering mail until space is freed.