  Given a lattice polygon $P$ with $g$ interior lattice points, we associate to
it 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. 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 $P$.
