It is well known that the set of possible degree sequences for a simple graph on $n$ vertices is the intersection of a lattice and a convex polytope. We show that the set of possible degree sequences for a simple $k$-uniform hypergraph on $n$ vertices is not the intersection of a lattice and a convex polytope for $k \geq 3$ and $n \geq k+13$. We also show an analogous nonconvexity result for the set of degree sequences of $k$-partite $k$-uniform hypergraphs and the generalized notion of $\lambda$-balanced $k$-uniform hypergraphs.