Yes, it is possible. I was without my telephone service for 5 days last week but my broadband worked okay.
I am not with British Telecom, but I've copied and pasted the following from their help/advice page for you:
///My phone's working but my internet connection isn't (or the reverse). Could this be due to a fault?
Yes. Calls and the internet make different demands on a phone line. A fault may stop one from working without affecting the other. If you're having problems with your broadband connection, however, you should visit our Broadband section for support.///