Suppose $X$ is a complete algebraic variety of dimension $n$. Must there exist an affine covering with $n+1$ pieces?

(For a projective variety in $\mathbf{P}^m$, we can always project it to some subspace $\mathbf{P}^n$ with $n$ equals the dimension of X, by a composition of projection from points. Since projection is affine morphism, we are done. For complete varieties, we have Chow lemma, thus $X$ is dominated by some projective variety, but can we find a such a covering or not?)