can someone disarm our ADT alarm system by removing the cable?
When an alarm trips, it can send out one or more of three types of alerts:


local - an audible alarm rings at the property
direct police connection - when the alarm rings, it goes off in the police station itself (rare)
central station - an alarm goes to the alarm monitoring company, which then alerts the police in certain circumstances (such as no one answers or fails to give a password following a callback to the house).


Some alarm setups are local only, some remote (police or central station) only, and some are both.

Two things control whether the alarm will go out:


is there power?
is there a connection to the monitor (if any)


Almost all serious alarms have battery backup to ensure that, even if the power is cut, the basic alarm system is still functional. If there is no battery backup, cutting power lines will nullify the alarm.

Local: Assuming power, AC or battery, a local alarm will ring upon intrusion, regardless of whether cables are cut.

Remote: Connections to police or central stations are most often handled through conventional telephone lines.  These may be landline connections or cable/internet connections (if phone service is via the cable system or other internet provider).

Even if power is cut, landline telephone service is often still active, since landlines have their own power feed. But if the telephone line is cut, the signal cannot go out.

Cable/internet phone, on the other hand, relies on modems that need AC power. If power is cut, cable phone does not work. If the cable line is cut, this would also prevent a signal going out.

Cellular remote: Some alarm systems offer a cellular connection. This is like having a dedicated cellphone built into the alarm system. These systems function even if the landline or cable line is cut. They will continue to function even if the AC power is cut (assuming a battery backup, which would be standard on a cellular type setup).

Whether your system would survive a wire attack depends on the type of system you have. Only a cellular based system with a battery backup can withstand all attacks on the wires that feed a property.  But many systems will detect a disruption of power or telephone connection, make a call to the owner to see if there is a problem, and alert authorities of the monitor does not get satisfactory assurances. The level of security is a function of how much you are willing to pay and how concerned you are about various levels of risk, and the particular protocols of the alarm monitoring service.