I have found that the BT operators are not terribly competent technically. I called to ask about a problem on my line. There was no dial tone, no broadband, no nothing, and I knew the rest of the village was of the phone network too. Gypsies had nicked 8km of telephone lines, this I knew for fact. I phoned BT, and Orange (my service provider) who both told me that there was no problem with the line, and that it must be a problem with the installation or equipment in my house.
I hope that you find someone who can advise you better than the BT operator, but I think if you do the basic fault finding as described I think that you can be pretty sure that you won't get billed by BT for any work done.