Yes. The set of $2^n$ (or $2^n-1$ excluding the empty set) dimensional vectors formed by entropies is called the entropic region [1]. Inequalities on the entropic region not implied by the nonnegativity of mutual information are called non-Shannon-type inequalities. The first such inequality for $n=4$ was given in:
[1] Zhen Zhang and Raymond W Yeung, "On characterization of entropy function via information inequalities", IEEE Trans. Inf. Theory 44, 4 (1998), pp. 1440-1452.
Since then, many more non-Shannon-type inequalities were discovered. Remarkably, there are infinitely many such inequalities even for $n=4$, as shown in:
[2] Frantisek Matúš, "Infinitely many information inequalities", in 2007 IEEE ISIT (2007), pp. 41-44.
Characterizing the entropic region is still a major open problem in information theory (even for $n=4$). The problem might even be undecidable depending on how you formulate it.