In a word, yes. Drum heads can split.
Drum heads are typically made from "mylar", a type of plastic. Mylar is pretty strong, but it is also flexible, allowing the drum head to resonate. Mylar will eventually wear out and possible split. It is difficult to tell when this will happen as many variables can contribute to this. Variables include: how much tension ins on the drum head (i.e. how tight it is), what kind of sticks you use, how hard you hit the drum, etc.
There are drums made from other materials. Most marching snares, for instance, are made from kevlar fabric. Kevlar, as you may know, is the same fabric that is used to make bullet proof vests. The popular Remo Fallams series of marching snare batter heads are made from kevlar. As you could imagine, these heads to not split nearly as often as their mylar brethren. Another popular marching snare head is the Remo Black Max/White Max drum head. The Max heads are made from Technora Aramid Fiber weave which is as durable as kevlar and acheives a superior sound. It is also easier on the wrists during prolonged practice sessions.