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: 
The only way a detection could be feasible is looking for changes in signal reflection. Neither resistance nor inductance will provide reliable information of bypassing activities, especially not with a mile long fence wire. However, with a sharp peaking signal (already present due to the nature of the electric fence) and analysis of the electrical reflection pattern might provide an usable information to detect bypassing. The term you want to look for is Time Domain Reflectometry. See: <https://en.wikipedia.org/wiki/Time-domain_reflectometry>
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.