Emails diverted to Drafts folder

Windows Mail: I'm seeing incoming mail going into the Drafts folder, and
outgoing mail saved to the Drafts folder. Not all mail, but a lot of it.
The same is happening in Windows 7 with Live Mail using a POP3 ISP
account, not Hotmail.

On the computers on which I've seen it, it's a relatively recent problem
that didn't exist before. Possibly related to some Windows Update?

I have some very irritated people badgering me about this and just can't
find an answer. Any light on the subject would be warmly welcomed.