The testing is using an automated tester within the main telephony switch. I performs several tests, one of which uses a Time Domain Reflectometer - simplistically a pulse is sent down the line and when it find a termination or damage it is partially reflected back, the time is measured and converted to distance and will give an approximate location of the fault. The estimated distance may have suggested it was within your property or close by - and the "It might cost" warning ensures you are aware that if the fault is on your side then you know a bill could be sent.
M H C
taurus excreta cerebrum vincit