It could certainly be true if you take your set theoretic axioms to be something other than ZFC. For instance I'm pretty sure that in ZF + AD (Axiom of Determinacy), every set of the real line is measurable.
EDIT: In fact, whether or not you like AD, it is the case that "every set is measurable" is consistent with ZF. Thus ZF (without Choice) certainly supports two models of R, one in which every set is measurable and another in which there exist non-measurable sets. I'm not actually sure how to interpret your comment "any two models of the reals are isomorphic".
