Do you have wooden posts, or metal ones? The wooden ones, while not a perfect insulator, are better than metal in that respect.
The fence needs to be disconnected from both terminals when you check this, since it provides a "load" for the system. Ideally, there will be very low resistance from the fence, and you'll have almost the same voltage at the end of the run as you do at the beginning. Every "short" to ground, such as blades of grass, brush, and what-have-you adds to the load on the system, and there is a corresponding drop in voltage.
When the horse, cow, whatever else touches the fence, there's a severe drop in voltage as their body absorbs the energy and they are shocked. The idea is that they won't tolerate that for long and will move. Some are more resistant than others to it, I've seen a man deliberately lay his forearm along the bare spark plug caps on a Chrysler industrial angine and stop it dead in it's tracks. He claimed to not feel "much" of a shock at all, while he was absorbing the energy from the ignition system. He sucked enough energy out of it to stop the spark plugs from firing.
This is what "shorts" do to an electric fence system, the inanimate objects such as grass, twigs , etc absorb power to the point that the "spark plugs", aka horses and cattle, stop receiving enough of a shock to deter them from getting into the fence. A "dead" short, such as metal contact direct to ground, (like a T-post touching a wire) is the same as many smaller shorts combined.
Not all fencers are created equal, as folks have mentioned. We have a four strand fence, two wire-embedded ropes and 2 steel 14 ga. wire strands around a 6-7 acre pasture. It's new this summer and has a ParMak 50 mile fencer powering it. No trouble getting well over 6000 volts on any of the 4 wires, even at the end of the runs. We use a 5-element fence tester that lights up each successive element to indicate how much voltage you have.
The temporary fencer we have is a Zareba 4-D cell battery job, the best it will do is a weak 2-element flash with the same fence connected on the same tester. I can grab the fence and tolerate the shock, but barely. If I was insulated with fur and REALLY wanted that patch of green grass I'd be ok with it I think.
Grabbing the wire with the AC fencer working properly is something else altogether. I don't care how green the grass is, that just HURTS!
If you're not getting a bright blue-white spark jumping a 1/16-1/8 inch gap across the terminals of an unloaded fencer, there's a problem with it. You may want to pull the cover off and have a look inside for corrosion or other obvious problems. It may simply be "done" after all those years...
Sean