Let M n be a biharmonic hypersurface with constant scalar curvature in a space form M n+1 (c). We show that M n has constant mean curvature if c > 0 and M n is minimal if c ≤ 0, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and Generalized Chen's conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space E n+1 or hyperbolic space H n+1 for n < 7.