Question: In the Flashpoint episode "The Farm," there's a scene where you can see one of the officers run a jumper cable from one contact on an electric fence to another, then cut the line of the fence (not breaking the circuit, mind you). Based on context clues, one could assume the fence surrounds a large, multi-acre plot of land. Another bit of information: the cable they used to keep the circuit complete was longer than the original cable on the circuit, which would mean it would have a higher resistance when the circuit was rebuilt.

I have two questions:

1. How feasible is this?
2. And would a (powerful, that is) fence controller be able to detect the change in resistance? Or would it be so small that it's undetectable in the grand scheme of things?

From my (limited) electronics/electrical schooling, adding the cable to jump between the contacts would change the resistance, as resistance for parallel circuits is modeled as:

$$r\_{total} = \frac 1 {\frac 1 {r\_{a}} + \frac 1 {r\_{b}}}$$

Which means the addition of the extra cable would have an impact on the resistance of the circuit itself (although very minimally).

If I recall correctly, another one of the officers was counting down to when the first officer should splice into the circuit. I don't know enough about electric fences to come to any conclusion, but are electric fences constantly charged? Or do they have a delay between pulses?

Answer: 
I am not going to show this to my family (we meet monthly at the family farm where there are multiple electric fences), because they would laugh at it. Electric fences are designed for maximum reliability when installed by someone with only the barest understanding of electricity. As such they have no components not needed except for an oversize case to protect it from irate cows. Further the traditional wire of choice has been iron which has high resistance, and rusts which increases resistance in unpredictable ways. These days the wire of choice in nylon carbon fiber mesh which in addition to having impossible to calculate resistance, the impedance and capacitance also are affected by the weather. To build an alarmed electric fence would require not only a entirely new class of "charger" (the box that charges the fence), but different wire, insulators, poles and grounding.
Some car engine controls look at wave forms in ignition secondary wires to learn if & how fuel-air mixture is burning. Some can spark again if burn is not as desired. A fence monitor could measure wave forms of voltage applied & current to detect fence changes. Measuring between pulses is not the only option.
Sorry I do not remember makes & models.