If you are able to make connections between the things you have learned, you learn better. So if know many different things, then you can make connections more easily. As a result the study of many things can aid in learning since connections come more easily. From that, I think it is possible for a polymath to exist, they could learn more quickly than somebody who learns one thing, and over time that increase in efficiency could add up into a proficiency in many areas. It would certainly take longer to get good in many areas than to be good in one area. Our society pushes us towards specialization because the modern education system has been made to prepare people to enter the work force.
It is at least true that a lot of people influential in individual fields have taken time to study many, and their influence usually comes from realizing something they studied from a different field applies to a problem they have in their first field. (Examples: Chomsky's Hierarchy of Grammar's influence on Computer Language theory, and automata theory. Chomsky studying Philosophy to formulate his ideas on cognitive linguistics, Karl Marx's study of economics to influence his sociological theories, Cognitive Behavioral Therapy being directly based on Stoic Philosophy, etcetera)
If you study what you find interesting, you won't get bored, so you will spend more time studying in general.
