As far as I can remember Modem technology stopped increasing their speeds at 56k because the phone lines themselves couldn't deliver data properly at higher speeds. Now we've got broadband on the way over the same lines, how can this speed increase be possible as the same limits should still apply as you're using the same phone line?

Quite simpy by bypassing the bit that limits the bandwidth.

It's not the lines themself that limit the bandwidth, it's the switching equipment in the exchange.
DSL overcomes this by using 2 different frequency ranges. The voice one travels through the exchange as normal. DSL traffic uses higher frequencies. A filter on each end of the phone line splits the 2 frequencies. In the exchange the DSL traffic is routed via a special card & bypasses the switching equipment.