Sub accounts fail to send emails?

I use AT&T DSL service and download all emails to a mail client, Mail (Macintosh). My main account can send and receive emails, but sub accounts cannot send emails, but they can receive emails. Whenever I attempt to send an email, a window pops up, saying that my sub account was rejected by server. All the settings are the same as the main account. Seemingly, when I send emails on the web, there is no such a problem. I am very puzzled. Any idea at all? Thank you.