Arresting gear wires rarely come apart. The wire is inspected regularly and before 100 traps is replaced. If there is a problem withthe wire or the motors, the wire can be removed quickly. If there was a problem with the arresting gear motors, them maybe that might happen, but more than likely the tail hook would come apart. The arresting wire is made up of more than one total wire. There are two sets of wires below the flight deck that are attached to very large pully and motor systems. The wires that come from below deck have two quick disconnects on the flight deck that attach to the wire you see on the flight deck. If there is a problem, they quickly disconnect the wire on the flight deck and use the other 2 or 3 arresting wires.
The tail hook is also removed before 100 traps and inspected. Tomcat and hornet tail hooks are made up of multiply pieces. The long tube that attaches directly to the aircraft and whats called a hookpoint which is what grabs the wire. The hook point is attached to rest of the tailhook by a large bolt. To have either an aresting wire or tail hook fail in the manner you ask about is rare these days.
If the wrong setting is set in the arresting gear pulleys and motors, one of two things will happen. If they put in too much weight, then the wire or tail hook could come apart, because the motors are trying to stop something that is supposed to be much heavier. If they don't put in enough weight, then the aircraft won't stop completely and go over the edge of the boat.
A good example of this is two rubberbands one is thick and one is thin. The thick one represents a high weight setting. It takes more force to pull it out. The heavier an aircraf the more force it has when it lands. The thin rubberband would be pulled out farther with the same force used on the thick rubberband.