The answer is yes, but you have to reference that child.

var target : GameObject;

function Start()
{
    target.BroadcastMessage("ClassToCall");
}

Yes, it's called SendMessageUpwards() :)

EDIT: Or did you mean laterally (eg: a child on the same level of the hierarchy?) Because BroadcastMessage should do that by default.

Depending on your other requirements, you could send the message from ChildOne's parent, or even its root (in case the hierarchy was many layers deep). e.g.

childOne.transform.parent.BroadcastMessage(...);

or

childOne.transform.root.BroadcastMessage(...);