I have just finished work on a large site with 135 apartments where i was involved in the comms system. Electrical contractor wired all BT's stuff, but then BT just came and terminated their own NTE5 master points and weren't interested in connecting up the cabling to bring the other six extension sockets into circuit. The people who move in just have the master socket in the lounge working. The lucky ones mention it at the right time, and I go and connect for them, (at no cost!), the unlucky ones just have a phone point in the lounge.
As johnny_t says you can connect the wiring yourselves, but if it produces a fault that you subsequently call BT out for, they WILL charge you. Undo the two small screws on the front plate, and half of it comes away. There are four connections on the back of this unit, numbered 2 3 4 & 5. Go and look at one of the extension sockets that doesn't work and note which colour goes to which connector. Buy a cheap disposable "Krone" type telephone insertion tool to push the sires into the appropriate connectors. DON'T be tempted to use a small screwdriver, it'll wreck the connections, and cause problems in the future.
This should sort your problem. If buy chance when you reconnect the socket the extension phones start to ring constantly, you've got the polarity wrong, so reverse 2 & 5.