BB *should* stay up but because it's one leg dis it will be erroring and so the modems will drop the speed in an attempt to correct. Once the line is fixed, it should go back to same speed it was before.
If it's BT (or sky/TT etc) ADSL then it will work with one leg of the pair disconnected.
Check your broadband speed, I bet it's down to a fraction of what you usually get.