Why "Classification" of 4 manifolds is NOT possible?

I'm guessing that you heard this from someone whose reasoning goes "Every finite presentation of a group can be made to give the $\pi_1$ of a smooth 4-manifold. If we could put any 4-manifold into the Magic List of All, then we could recognize presentations of the trivial group. But no algorithm can do that."

Often people worry about classifications of simply connected manifolds, and don't have to deal with this. (Of course in three dimensions this becomes Perelman's theorem.)

As pointed out in a comment by Richard Kent to Allen Knutson's answer, the problem is a bit more subtle than it may appear. In order to prove that the homeomorphism problem for compact 4-manifolds, say in the topological category, is recursively unsolvable, it is not enough to know that (1) every finitely presented group can be realized as the fundamental group of some compact 4-manifold, and (2) the isomorphism problem for finitely presented groups is recursively unsolvable.

Instead, what you do is give a construction which to any finite presentation $< S | P >$ of a group associates a 4-manifold $M(S,P)$ in such a way that $\pi_1(M(S,P))$ is isomorphic to the group defined by the presentation $< S | P >$, and moreover two such manifolds are homeomorphic if and only if they have isomorphic fundamental groups.

Then you have constructed a class of 4-manifolds for which the homeomorphism problem is equivalent to the isomorphism problem for finitely presented groups, and therefore unsolvable.

About "geometrization for manifolds of dimension 4 or higher", well as far as I know there is no theorem which says it is impossible. It depends on what you mean by `geometric structure', and what you want those structures to do for you.