The alarm company might notice a 'problem' with the line eventually, but they would not know if it was an intruder cutting it. If you are really serious about your home security, find or make a system with an alert base on a 'mobile phone' (SMS) and back-up battery.
I used to have a monitored alarm, got set off accidentally once, they tried ringing all 3 keyholders couldn't get hold of them so called the police. Police arrived 90 minutes later, so the point of a monitored alarm is.....
Went out and bought a dummy box with a light for �20.00 instead
The gardner cut through our phone line that was connected to the alarm..... and the police turned up straight away. The company dealing with the alarm obviously got a signal that the line had been cut
I contacted LaserShield, and it seems they will offer Cyclone soon.. Cyclone supposevely communicates nationwide using the GSM cellular network described by smssms. This is a additional cost of 229.00 which bumps montitoring to 29.99 per month. So,,,, that would be 430.00 for the total system. The Cyclone system is not yet available, and those considering this option are asked to contact them to be put on a first come, first serve waiting list..
I used to be a shift manager in an alarm monitoring company. It depends what sort of alarm system you have. If you have a standard "digi" or "dialler" then there is no wat that the central station can know that the line has been cut. Some systems such as BT redcare will monitor the line. If there is an interruption in the signal - the central station will try to "up" the line - or communicate with the device. If this is unsucessful then they should take appropriate action.
The cellular type systems have been done before - when I was at the central station (many years ago) we trialled a system called "paknet" that worked on a mobile signal - it was not very sucessful. I would ask your installer about a redcare system or similar.