This is an Ongoing problem with the built in Mac Mail program. Almost everyday there is at least one post, if not 10 posts, about the same thing on these forums.
Most Mac Fan Boys will respond with "It must be your ISP has changed settings". But every other device you have, Phone, Pad, other computers, whether Mac or PC, have NO problems getting and sending mail with those exact same settings as used in Mac Mail and that Mac Mail worked a few hours ago or yesterday and now it doesn't.
I suggest you switch to some other Mail program, Any other Mail program, and this type of problem will never happen again.
I have a similar problem where all of a sudden Mail stopped sending. I no longer hear the sound of sent mail and nothing happens. It just started a day ago on my laptop but not on my desktop. I've done all sorts of things like fixing disc utilities and all the troubleshooting of the article Linc Davis posted.
One idea is that I was exporting mailboxes to the laptop the other day and it seems that perhaps something with that action stopped the sending function since it's after that that this problem has arisen.
Any ideas here?