In addition to Andreas' remark on the number of non-isomorphic models, perhaps it's noteworthy to say that many non-structure theorems are aiming at the construction of many models which not only are non-isomorphic but also are hard to tell apart. This may be achieved by requiring the models to satisfy the same set of sentences of some infinitary logic. By taking infinitary logics into consideration, we may consider a theory T to be classifiable if elementary equivalence in L_(infty, lambda) is a sufficient condition for an isomorphism between a given pair of models of cardinality lambda (this should be found in Shelah's book).