Actually, depending on the Telco network you're being fed from, many lines can indeed be remotely checked with line test cards although I doubt if a call centre had the management platforms to do that.....
Anyhow, firstly borrow another phone and try it, if that doesn't cure the problem locate your main incoming socket. If it's got a 'split' front face your laughing. Carefully unscrew the two screws and pull out the half face. This will disconnect your extention sockets. All connections will possibly be made but you actually only need three wires. Disconnect the wiring connected to connections 2, 3 and 5. Substitute 3 of the other wires and re-connect them but make sure that you replicate the same colour wires to the numbered connections at both ends of the wiring i.e. at the extention socket too. If this doesn't cure the problem you've got either knackered wiring (it sounds like it is getting wet - does it run externally to the house?) or a knackered socket. Either way it's BT's to fix.