ban-all-sheds said:They do as far as BT are concerned
Not in terms of your credit score.
However, obviously if there was a debt to BT from the previous occupant, BT will ask for proof that you are in fact "ban-all-sheds" and not "dodgy-bloke-who-used-to-live-in-ban-all-sheds-house-and-owes-us-money"! Normally any proof of address with your name on it will do. Otherwise anyone could run up a phone bill and then ring BT claiming to be a new occupant.
This only occurs when the debt is recent. If the debt is years old, BT will see from the credit scan that the debtor is no longer on the electoral roll and that you are (if you have been registered long enough).
As long as you are now on the roll, the misdemeanors of the previous occupants will not affect your credit rating at all. As mentioned to Adam, if you want confirmation, email me your details and I'll run it through the Equifax system and pass on the report.