Given a lattice polygon P with g interior lattice points, we can associate to \(P\) two moduli spaces: the moduli space of algebraic curves that are non-degenerate with respect to \(P\) and the moduli space of tropical curves of genus g with Newton polygon P. We completely classify the possible dimensions such a moduli space can have in the tropical case. For non-hyperelliptic polygons, the dimension must be between g and \(2g+1\) and can take on any integer value in this range, with exceptions only in the cases of genus 3, 4, and 7. We provide a similar result for hyperelliptic polygons, for which the range of dimensions is from g to \(2g-1\). In the case of non-hyperelliptic polygons, our results also hold for the moduli space of algebraic curves that are non-degenerate with respect to P.