My friend is having the same problems, Lona. He has also been getting random disconnections since the Pipex upgrade. They sent him a whole list of things to do, by way of checking all his hardware (filters, modem, etc), and at the bottom of the list, the BT Line Test you mentioned before. I am still a little confused as to how that works. How long are you supposed to leave it running? If it disconnects, are you supposed to reconnect, and if so, how many times should you do this? Where does the information go, to BT or to Pipex, and how is this followed up?
I have to agree with you, in that it does seem that Pipex should be sorting this out with BT. You are paying Pipex for the service, which includes support, so if there are any problems, Pipex should be sorting them out, not passing the buck to the customer. My friend has given up, for the moment, because it's too technical for him, and he honestly can't be arsed to go through all their suggestions, when the fault is clearly something to do with the upgrade.
I would 'do your worst' with them on the phone, Lona, and threaten them with further action if they don't sort things out. Even if the fault lies with BT, as I said, you are paying Pipex, so it's down to Pipex to resolve, just as if you bought a cooker from Currys, the onus would be on them to have it repaired if it went wrong, not on you to deal with the manufacturers.
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I will reserve judgement on my current situation with Pipex, but I asked them if they could transfer my service to my new address when I move. They said no, I would have to cancel the existing service, giving 30 days notice, then sign up again once I have moved, as I can't have two accounts running on the same phone number. As I am moving in just over two weeks, and am taking my phone number with me, if they insist on the 30 days cancellation term, I will therefore be without service for two weeks from moving in, plus however long it takes to initiate a new account. When I said to them that I would therefore be better off using another broadband provider, they said they would "see what they could do" about the 30 day term, as I have been with them for three years, and wished to continue with the service once I move. I sarcastically replied "thanks very much", and I await their further communication.