I had EXACTLY the same problem a few weeks back.

Turned out it was a buggered piece of copper in the distribution point (box) at the end of the road.

Took a lot of back-and-forth between Myself, ISP & BT. Zen Technical Support were great, BT needed serious motivating, which Zen took care of for me. I resisted going through to BT direct, even though I had noise on the voice line, preferring instead to let Zen deal with them.