The process the OP describes is sometimes called "reiterated mashing," and it does indeed work--though as others have noted, efficiency is less than optimal.
This past fall, I made a wort of about 1.037 (pre-boil) and performed a second mash of equal size in it, netting another 30 points. I don't have my notes with me, but after boiling, my gravity was at least 1.080.
I did it because I am a stovetop BIAB'er, and none of my vessels are big enough to mash 10 lbs of grain at once. Reiterated mashing solved that problem, but I don't think I'll try it again anytime soon. The beer I made in that instance was just too much--ABV too high, crazy mashing protocol, too many additions of spice, orange peel, sugar, oak... I learned that I enjoy brewing simpler beers, and I enjoy drinking smaller beers.
But to the OP, if you want to give reiterated mashing a try, it is at least an interesting experiment. There was a good discussion of it on the 11/22/07 episode of Basic Brewing Radio, as well as in "Radical Brewing."