C’mon, this is the Straight Dope. Let’s put some muscle into this analysis, shall we? The following is not a proof, but is data that happens to be borne out by my own experience with knives, spears, arrows, etc.
Hand thrown wooden spears impacting at 30 or less mph can easily pierce flesh with lethal results. Atlatl thrown darts (which are shorter, lighter, more flexible wood pieces, unlike spears) can impale game at 30-40m (100-130 ft) with an initial velocity of up to 100 mph (160kph) The pitcher’s mound, first base and third base are all within atlatl dart range of home plate. Second base is a bit too far for a standard atlatl throw, but within the range of the longest recorded throws (and we have very few, unselected atlatl throwers these days - much less selected and less thoroughly trained than than Major League power hitters)
If everything is just right a bat could act as a crude two handed atlatl, wielded by a highly selected, highly trained swing maximizer.
A ball struck by a baseball bat can acquire considerably more velocity than a pitcher’s 100mph pitch (A batter can hit a ball up into the far centerfield stands. A pitcher at home plate can’t come close to throwing a ball into the stands - outfielders can manage about half the full field length in a low arc, and practive throwing a ball infield towards a runner coming home.)
It is not unthinkable that a small fragment of a bat could approach the velocity of a well-hit ball, and certainly the much lower velocity of an atlatl-thrown dart, even after some energy was absorbed in breaking the bat (which would usually absorb a large amount of the energy, some of which might be returned by the complex kinematics of the flexion and elasticity of the shattered bat). It is somewhat more likely that a fragment or large splinter might achieve the lower still impact velocity of a wooden spear or a (fairly light and flimsy) wooden blowgun dart.
The fragment would, however, be very unlikely to have a straight flight with no tumble. In addition to its energy and momentum, it would likely have a considerable angular momentum (tumble) about the centrum where it broke from he bat. Aerodynamics and the equipartion of forces and momenum between several fragments, along with the phase of the moon and an ancient gypsy curse might, however line up he forces just right. Also, thrown knives, which are admittedly harder, denser and sharper than wood fragments (but are also thrown at much lower velocity) routinely impale themselves in hard wooden targets after a tumbling flight at fairly low impact velocities
I would conclude that it would not be absolutely impossible, but it would be an extremely freakish occurence.
Some links on atlatls and atlatl darts:
American Journal of Physics
(abstract only for the public; full text available to subscribers)
From New Scientist 15 May 1999 pp40-43
As I said, this is not a proof by any stretch, but it would seem that the impact would contain more than enough energy to launch a sharp splinter of a bat, weighing less than the few ounces of an atlatl-thrown dart, into an extremely unlucky player. People have impaled themselves on not very sharp branches at very low velocities (i.e. tripping and falling into freshly trimmed bushes); the impact profile is admittedly very different, but it does serve to illustrate that the sharpness (or hardness) or a spear or arrow head is not absolutely necessary ot penetrate skin. Once the skin is penetrated, the internal organs can be pushed aside easily (as some of us know personally from performing abdominal surgeries)
I wouldn’t expect to see it happen in my lifetime, but I would not be utterly stunned if it did, either. Certainly, it would not be difficult to (as our mothers might have said) “put an eye out’ and the bone behind the eye is paper thin and remarkably easy to shatter - medical students do it by accident all the time in the lab. The temporal bone (the ‘nearly impaled temple’ one poster alluded to) is also very thin, and has been shattered by some rather surprisingly small impacts. I trust the potential lethality of a 3” splinter in the brain (and more importantly in the delicate vascular structures (blood vessels) in the cranium) are self-evident, even if it’s exceedingly unlikely to happen