I’ve always tried to discourage people from emailing files bigger than 10MB, even if their email host says they can. Email just wasn’t built for it. Turns out, Apple Mail wants to discourage this behavior, too. Or, more likely, it just has a massive freaking bug that should have been addressed years ago.
The bug is, if you try to send an email with an attachment that’s too big for your host, Mail.app can’t get the message out the door, and it stays stuck like Winnie the Pooh in Rabbit’s door, blocking anything from coming in or out.

To continue to mix my animal metaphors, this bug coincidentally randomly bit four different clients in the same week, so I figured I’d post the solution:
- Quit Mail.
- Go to ~/Library/Mail/Mailboxes.
- Move Outbox.mbox to the Trash.
- Open Mail again.
That should fix it.