I've had this problem with BT internet using Outlook as the mail client. In fact I downloaded Thunderbirsd last night and reset my BT password to see if a different mail client would solve my problem. It receives OK but will not send email, presenting me with the same message you get.
In Outlook I get either a request for a network password or a POP3 error saying the outgoing server is not recognised. When hitting the Test settings button the test email won't go for the same reason.
Initially I took this to be an Outlook problem and I used Windows system restore to go back to a point where I knew the email was OK. This worked initially but has now reverted to the above situation.
My final thing is that I have created a new email account with Bluebottle which I will try with Thunderbird tonight. If this works I can only assume the problem is with BT. I'll let you know what happens.
Incidentally I agree with Groundgripper that the internet service is unreliable. I went through a period recently where I could not access the internet at all anytime after about 16.30 gmt. In the daytime no problem. I spent hours on the phone with BT on this and thwy were not at all helpful. I couldn't understand it as we have a separate ADSL socket. What eventually worked was to unscrew the socket from the wall and plug the router into the test socket that is inside the box. I haven't had any problems since then.