In short, yes.
But some ISPs make it simple for you (going so far as to let BT swap your broadband when they swap your phone line) and others make it much more difficult.
I'm with PIPEX and have enjoyed a great broadband service for a couple of years, but when I moved 100 yards up the road I ended up without broadband for 2 weeks
So ask this explicitly before you sign up with your ISP of choice.
RC