I'm having precisely the same problem.
In my case it's the handset on a fax machine that's playing up - I get sporadic interference whenever the broadband router is running on the same line. Like you, I've tried everything I can think of, but nothing has worked.
Let's hope we both get an answer. I suspect that my problem is related to the fax machine - it's new, and when we had the old one everything was fine.