My father had exactly the same problem which we initially atributed to ISP or BT problems, but, to avoid the long wait in (and expense of!) call queues on technical support lines and then the inevtiable ping pong between companies....
I bought him a new modem & router. Plugged them in and they refused to connect, no matter what I did with the config. So I tried the new modem with the old router. Connected but dropped out in exactly the way you describe. Then tried the old modem with new router, it connected first time and has remained so for over two months now.
Conclusion? I don't really know - but try replacing your modem/router before blaming your ISP!