If you turn the router off (assuming you have one), then whilst listening through the phone handset, turn the router on and see if you can hear the router connecting (it would sound a bit like an old fashioned dial-up noise).
If you can hear it then you have a problem, the frequency the broadband comes in on should be out of the audible range to stop it intefering with the voice calls, but problems with the external wiring, or at the exchange can cause a shift.
If a broadband engineer came to see you, one of the things they would do is a 'quiet line test'. You can do this yourself - dial 17070, then choose option 2. If all is fine, you should hear absolutely nothing during the test.
I know this might not solve your issue but it might help you if you have to explain what is happening to someone else.