Sky provide your broadband so Sky are responsible for sorting it out. After all, you pay them for it don't you?
They should liaise with BT Wholesale or Openreach to get it sorted. They will often tell customers to call BT and get the line tested because it saves them paying Openreach the (minimal) cost for a line test.
BT provide you with a telephone service only. If a BT line test comes back as OK that just means it can't find a fault which will affect your telephony service. There may well be something on the line causing a broadband problem however.
If you call a BT engineer out to look at the problem and he/she finds a fault on the BT line then the fault is fixed free-of-charge of course. However, if the engineer finds that the problem has been caused by your own equipment connected to the line such as extension wiring, modems, 'phones etc - or you have damaged the BT line or main socket - then the engineer will fix the problem and you would be charged £129.99 for the repair.
Go to
www.youtube.com/user/btcare for some videos on how to do some basic checks at home first. Providing your line speed is fast enough of course!