  Let $M^n$ be a biharmonic hypersurface with constant scalar curvature in a
space form $\mathbb M^{n+1}(c)$. We show that $M^n$ has constant mean curvature
if $c>0$ and $M^n$ is minimal if $c\leq0$, 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 $\mathbb E^{n+1}$ or hyperbolic space $\mathbb H^{n+1}$ for
$n<7$.
