Most of the modern electronic meggers will deliver a maximum of a few milliamps into a short-circuit. The current likely to flow through your body if you grab hold of the probes will obviously be less due to your body's resistance.

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People tell me "It's the amp load that will kill you, not the voltage".
If I say, "Okay then, grab onto the hot lead of 4160", they say, "Well...at high voltages, the voltage alone will kill you."

Can somebody explain this?

It's not the high voltage in itself which would kill you, but rather the high current flow which results from that high voltage.

It is the current which does the damage during an electric shock. How much harm is done depends upon the intensity of the current, which parts of the body it flows through, and for how long. As the current and the duration are increased, so there is more chance of burns. Current passing through the vital organs of the body (e.g. the heart) is far more dangerous than that passing between, say, two fingers on the same hand.

Ohm's Law applies to the human body just as to metallic conductors. Thus at higher voltages you will get more current for any given value of body resistance.