I was thinking of something where they put their heads down and just ran. The pikes would be easy to ward off because of the point of contact being close to the shield and far away from the pike-wielder (i.e. simple leverage).

I would have thought that this could happen pretty quickly - a phalanx (to my way of thinking) would have been extremely effective if a group of men tried to push forward against it with a horizontal shield wall (as the shields would be perpendicular to the spears) but not trying to his shields obliquely.