Standard Operating Procedure for customer-facing enterprises in Britain. BT must train its staff in the same school as Cooperative Energy! They borked our dual-fuel switch (transcribing the wrong meter ref.no.) and since Sept 2014 have been (double) charging us for electricity they never supplied!
Sounds like you're in the same Monty Pythonesque situation. "You must be receiving broadband from us, as you've been paying our bills!"