We assume that all solutions of a two-dimensional, periodically forced differential system (of period T) can be continued for all future time. If there exists one solution that is future bounded, then there exists a solution of period T (Theorem 3.4). This is the Massera theorem. To extend the Massera theorem, we assume that there exists a future bounded solution that is also bounded away from a known T-periodic solution ζ. We prove that either there is another periodic solution of period qT for some integer q ≥ 1 or all compact motions that remain a finite distance from ζ have a well-defined irrational rotation number about ζ (Theorem 4.3).