Sounds like a problem with your BT line. I had similar problems a few years ago on dialup. After a few visits from BT, their 'engineers' telling me that it was my equipment that was faulty, a big argument with BT over their charging me for a fault which they hadn't cured, eventually they sent somebody who knew what he was doing. He plugged a phone into the test socket & heard noise on the line. He said: 'I'll just be a few minutes -I need to go to the box at the end of the street - don't try to use the phone until I go back, as it may go dead'. A few minutes later he came back & said that the noise was audible there, so he'd connected our line to a different exchange line, but he'd need to connect it at the exchange before the line was usable, which would take about a quarter of an hour. Ten minutes or so he rang from the exchange to tell us that the job was done & to confirm that we now had a noisefree line, which we had.
The moral of this story? To get good service out of BT you've got to persevere!