ping mail.btinternet.com
If you get four "Reply from 217.12.13.134: etc... " its probably a port issue, since the name has resolved to its IP address. If you get
"Ping request could not find host mail.btinternet.com etc, " then you've probably got a DNS fault, i.e. your router's lost contact with the BT Domain name servers or is getting the wrong ip address from them for the mail servers. You could also try pinging the bt mail server direct using the Ip address I found, i.e.
ping 217.12.13.134
if you get replies for that then the network route to the servers is probably OK so its almost definitely either DNS or a network port blocking fault, if you don't get replies then you have routing error outside in BT land someplace..
