Outgoing shouldn't make any difference as it goes via BIS and the carrier, not through your ISP directly.

And what do you mean "BIS isn't getting a response"?

The problem must on the incoming side. Are you setting it up via POP or IMAP? Do you have "leave mail on server" checked in T-bird's set up?

Have you talked to the hosting company and made sure you have the correct settings and if they support BB.