Most e-mail clients give you the option of authenticating or not authenticating when sending mail. Depending upon how your client is configured, you might be able to receive e-mail, but not send it, or you may only be able to send to other addresses at your domain. The problem you've described really seems to me to be a symptom of Thunderbird not being configured to authenticate when sending e-mail.
Try this. Under the Tools menu, select "Account Settings..." and go to the configuration pane for Outgoing Server (SMTP). Is the checkbox for "Username and Password" checked or unchecked? I believe it's supposed to be checked.
I've been experiencing some strange e-mail problems lately as well, which I think have been related to problems with authenticating -- the difference being that users at my domain could send e-mail fine, they just couldn't receive it. Weird.