I experienced this problem on both my laptop - wifi connection and my desktop - direct connection. BT customer with a Hub 5 for over a year this started after I upgraded to unlimited from 20GB a month. Computers connect to the hub ok but the hub says "no internet access" however my BT TV Vision box was still working showing BT Sport so that was connected to the internet.
Spent most of a Sunday on the helpline trying various things but nothing worked except turning the hub on and off which worked occasionally.
Eventually mybt account website said a problem at the exchange and the problem has occurred less and less and has now stopped...(crossing fingers)