Try using the Connection Doctor diagnostics to confirm the correct operations of the POP or IMAP server you're using. (Mail.app > Window > Connection Doctor)
If there's a problem somewhere with the Mail.app "receive" path, the Connection Doctor diagnostic tool will usually report it.
If the diagnostics pass, try sending a message to the failing account from a web mail client, or another working email account.
Also check with whichever account provider you're working with; many have status displays or web clients available, and these can be used to determine if there are issues.