Succeed in which sense? Take the traveling salesman problem. You can't analytically solve it for N in the hundreds. But there are heuristics which give acceptable results and are widely used in practice.

It depends what you mean by acceptable. For many problems, assuming that P != NP, one cannot get approximations beyond a certain amount. In particular, one cannot get approximations which asymptotically approach the optimal. See for more details my earlier comment here.