BOAC,
I'm not 100% (only 99.9%) on the BT angle (I work for a different telco), but unless you are with a cable company then BT owns the lines between your house & the exchange. Some ISPs only rent you the time on their network and you have to rent the line itself from BT, and some ISPs have done a deal with BT to pay the line rental on your behalf (you'll see BT OpenReach which is the bit of BT that supports this). Either way - the end company responsible for the wires between the exchange and your house is BT, that's why they get to do the diagnostics.
As an aside you'll see items in the news whereby BT are having to allow ISPs to install their own equipment at the exchanges at some point real soon. This is why they are setting up OpenReach.
As for the failure - any high speed links like your broadband between your house and BT are still running down the old copper wires that have been there since time began (even new properties have the same quality telephone cable). However the physical signal demands of high speed working are a lot more susceptible to tiny changes in line quality unlike the voice lines which can take a lot more abuse. It could be that the dampness is affecting the noise characteristics of the line from time to time which is why your broadband goes for a hike.
I hope you get it fixed soon