Pick receiver up and can start conversation provided caller hasn't hung up.

Can dial out calls ok

Broadband is fine

What's likely to be wrong?

Any ideas before I start disconnecting things, and calling BT?

I would try removing the master socket faceplate and connecting one phone at a time dirctly to the test socket underneath. This cuts out all of your extension wiring, filters ADSL etc, and also allows each phone to be tested in isolation, assuming they are all easily portable.

It's possible that only one faulty phone is overloading the ringer circuit.

If the fault is still there for every phone it's most likely down to BT.

If all phones ring OK when tested at the master socket take a close look at the internal extension wiring, replace the master faceplate and start again with only one telephone connected at an extension.

Check this BT link out on proving problems on your telephone line.
http://bt.custhelp.com/cgi-bin/bt.cf...45%2C353%2C374
Your fault is commonly known as "ring trip"

I had exactly that fault last year - turned out to be some old extension wiring to another socket where the insulation had broken down.