A rainbow q-coloring of a k-uniform hypergraph is a q-coloring of the vertex set such that every hyperedge contains all q colors. We prove that given a rainbow (k - 2left perpendicular root kright perpendicular)-colorable k-uniform hypergraph, it is NP-hard to find a normal 2-coloring. Previously, this was only known for rainbow left perpendiculark/2right perpendicular-colorable hypergraphs (Guruswami and Lee, SODA 2015). We also study a generalization which we call rainbow (q; p)-coloring, defined as a coloring using q colors such that every hyperedge contains at least p colors. We prove that given a rainbow (k - left perpendicular root kcright perpendicular; k - left perpendicular3 root kcright perpendicular)-colorable k uniform hypergraph, it is NP-hard to find a normal c-coloring for any c = o(k). The proof of our second result relies on two combinatorial theorems. One of the theorems was proved by Sarkaria (J. Comb. Theory, Ser. B 1990) using topological methods and the other theorem we prove using a generalized BorsukUlam theorem.