They can't force you to pay the bills--that is, they cannot sue you for the money. But a private company can refuse service to someone if a close family member failed to pay her bills, the same same way you could look for a different service provider if a family member of yours had a bad experience with a company. No law forces you and them to do business together. So while they can't sue you for the money or otherwise legally compel you to pay, it is legal for them to say that if you want to do business with them, you have to clean up your mother's past due account.