BOAC,
We used to have a similar problem with the BT connection - although this was before broadband. Whenever there was heavy rain, the line detiorated or dropped entirely. Was enormously frustrating with dial-up internet access being the first to fail.
Only persistent complaints did any good - eventually our incoming copper line and termination box were replaced. Problem seems to be solved.
The legacy - I swore blind to have nothing to do with BT again, which is why I have cable broadband.
SD