I have found that the method of lines is a very natural way to think about the discretization of PDE's. Therefore I always default to that mindset when presented with a new set of equations. I have never seen a PDE where this would not work.
What I am wondering is if there are discretization methods (or types of PDEs) which can not be formulated through method of lines. I expect that any PDE where the time derivative is implicit in the equation and can't be solved for would be one such case (although I know of no actual example of this). I am looking for reasoning as to why the method of lines is always applicable or a counter example.