It looks like potentially a routing or peering issue may be the problem as since people can often get to some sites it rules out their connection. The clear dates listed on the BT service page range from 28th June to 1st July if the fault listed is the same one. So while we do not know the exact problem yet, this post if BT users can read it will at least save them pulling their hair out thinking it is their computer/router at fault.
Update Saturday 11:21am BT users on our forums are reporting that websites are becoming available again, so the problem may be over for now. First report we have was at 9:53am.
"We can confirm some customers had issues connecting to the internet on Saturday morning. The issue was fixed within a couple of hours and we apologise for any inconvenience caused."BT spokesperson on Saturday morning incident
Update Monday 30th There is the standard statement from BT above, but we have asked for more detail and so far all that is being revealed is that is was a network incident and not a failure of the filter/parental controls system that some suspected.