The problem with this is that the King has absolute rule. He can, theoretically, kick you out or carve up your lands at a whim. Obviously, there is a vested interest for a King to NOT do this as he often needs support from those same Houses, but in the prime Targaryen era with dragons, they kinda can do WTF they want.
So trying to play neutral will just get you attacked by one side or the other and cut off any support from the Crown. Kings often need lands and titles to dole out to supporters so if they see a House playing both sides that's the place to threaten in order to get their direct support. Barring a vast geographic distance (like the Starks and the North in general) most Houses can't sit things out for long lest they get hit themselves.