"Possible", "necessary" and "real" are the three modes of existence in ontologies with possible words. One of the main possible-worlds models has been introduced by Leibniz. But even in these ontologies, "possible" does not imply "necessary". Instead: "Possible" is defined as "real in at least one world". While "necessary" means "real in all worlds".
I would not complicate the original question with the concept of infinite time. At least, the concept of infinite time opens up new questions ...