iCloud: If an iCloud mailbox gets full
If you use the Mail app on your Mac to receive iCloud Mail and your iCloud storage fills up, you won’t be able to receive new mail until you make more iCloud storage space available. For more information see:
Mail on your Mac stores email in iCloud because by default Mail uses IMAP for your iCloud email account.
For more information about using Mail on your Mac, open Mail, then choose Mail > Mail Help.