Why such a dramatic drop in power when the handlebars are lower than the seat? I have done long rides with the bar 10+ cm below the saddle without getting a sore back, or anything.  Thing is, when I do, my average speed is 2 - 4 km/h SLOWER than when I have the bar higher up.  

I do know you lose power as you get lower.  Weird thing with me, is that if the bar is lower than about 4 cm below the seat -- and I mean by just 5-10 mm -- my speed/power drops like a freakin' rock!  (I'm talking as much as 3+ km/h slower up a 10% grade!)  Any idea why such a dramatic drop in power?

My bar is only 53 cm from the seat, which seems awfully short to me, for someone who is 186 cm tall.  (If I increase that reach just another 5-10 mm, I also have that same sharp power drop.)  I do know my torso is shorter than normal, given my 91 cm inseam, but still.  I think I have reasonable flexibility -- I can touch the floor with my fingers while bending over.  I do know when I had the seat lower and/or further forward, that caused additional power loss when getting low, but I think I have the seat figured out now.

UPDATE:  Today I fooled around -- yet, again!!! -- some more.  Discovered that if I concentrate on flattening my back while leaning forward, I can actually go FASTER than the more "relaxed" position (when the sudden power loss occurred).  (Sorry, not too familiar with anatomy . . . that's the best I can describe it.)  As noted in one of the responses, this "leaning forward" motion did require moving the seat 5-10 mm forward to get to what seems to be optimal.  I'll see how the "new and improved" position works tomorrow.
I agree with the biomechanical hypothesis already mentioned by @DanielRHicks.

Every joint/muscle-group has an optimal range of motion and an optimal max-force/max-power position.

I am the same height as you, and the distance you mention is pretty short indeed, as measured on the bikes I ride.

There seems to be necessary to analyze the "cockpit triangle" of your bike: bottom-bracket/saddle/handlebars: in the hypothesis that, lowering your bar, you get an angle between torso and thighs that is too acute, then the correction of the problem would be opening that angle.

That could be made in two ways:


Rise the handlebar (which apparently is the opposite of what you want to do);
Move all your upper body front by putting a longer stem AND moving the saddle forward a bit, so that you feel the pedals "more behind" than in the current position.


As observations:


High power output (a.k.a FAST) bikes, with aggressive riding position, tend to throw the rider over the bars, so that the torso weight can counter-act the torque on the pedals, and also be more aerodynamic. It is common to see such bikes with high, forwarded and sloped seats (extreme example: time trial bikes);
If you just put a longer stem without bringing the saddle forward, you get in a more "superman" position, actually DECREASING the torso-thigh angle that we would want to increase.


Finally, there are mysteries in finding the perfect position on a bike. Perhaps even sub-conscious factors of varied nature might contribute for a sensation of "not-right". Unfortunately, only self-analysis and trial-and-error are the guaranteed key to salvation in this case, although some professional advice would be always welcome (a.k.a. bike-fit).

Hope this helps!