Just to confirm this is probably far from uncommon, but the bit about "we had been advised repeatedly that we would have to pay if the fault was inside the house" is perfectly true, and something ISPs feel they have to warn about, and the reason they go painstakingly through all sorts of boring scripts to get users to check everything three times (OK, OK, but you know what I mean!)
When I moved from one area to another in N Wales to a home where the line had not been used for some 7 years, the first thing I noticed was a hum on the line, and I complained, even before broadband was set up. Engineer was unimpressed, sat outside and had his lunch then checked back some of the way (I would say 'all' but I would not be surprised if he didn't do much more than he felt necessary) to the exchange. Came back and suggested I'd just have to "put up with it" (the hum).
I had 6 months of hassle with that line. Back in 2001, I was getting 500 kbps (that was the standard speed for ADSL broadband), but there I was, in 2010, lucky to get 150 / 160 kbps (too slow to do anything with iPlayer but listen to radio).
When it rained (and we had a few showers in that summer), I lost not only the broadband but even voice calls for up to 5 days.
Luckily, I also had a Three broadband dongle, so I could run the BT line test online from my laptop that way. That kept on pointing the finger at the house wiring, and because of the possibility of a hefty fee (I think it was 100, but now 125 quid) I didn't request any engineer visit.
(Incidentally, the cheeky sod who told me I'd have to put up with the hum classified the problem as house wiring, and there was that big call-out fee on a bill about 2 months later, which I complained about, and had BT refund to me.)
Anyway, finally, when I could stand it no more (I think it was August or September and the line was activated in early February 2010), a young engineer came on Saturday, off his normal Merseyside patch, and by the time I got to the door (9am, and I was more asleep than awake) he was already up a ladder, because just as for the OP, the problem was perished insulation on the dropwire from the gutter level into the hall/ master socket. He spotted it from the garden and instantly knew the solution!
Since it was an old house, and old wire, and old master socket, and I had my computers up in one of the bedrooms, I asked if he could fit a new socket and new wire in that bedroom... Speed went from 150 kbps to 2500 kbps (2.5 Mbps).
Now I am on Merseyside and get 10+ Mbps on both my phone lines, but hope to go to 35 Mbps soon... no VirginMedia on my road, and the chippy behind me is meant to be able to get 70 Mbps (from a different cabinet).
Anyway, the 300 Mbps BT service is about 1,500 quid to install and 100/month wholesale, so no idea what an ISP would try to charge me... will have to wait until my business takes off :)