A couple of years ago I watched an episode of Mythbusters where they "proved" that you cannot split one arrow with another arrow.
The next day, I watched a show on the History Channel where an expert archer did exactly what Mythbusters claimed could not be done.
Well, first of all, they were investigating the myth that Robin Hood (or his non-fictional, 12th-century equivalent) could have split an arrow. So off the bat they were limited to the kinds of arrows a medieval archer would have on hand. They acknowledged that you could split arrows made of bamboo or composite materials, if I remember right. But of course Robin wouldn't have had those materials on hand: his arrows were made of wood.
The biggest problem they discovered with wooden arrows was that they tend to split along the grain, so if it isn't parallel to the length of the arrow, then the arrow doing the splitting won't make it all the way up the length of the shaft. I would assume from that, that an arrow with just the right grain could be split.
I was actually surprised that they didn't give a verdict of "plausible" for this myth (under the right conditions, however improbable).