  We prove rigidity results for compact Riemannian manifolds in the spirit of
Tachibana. For example, we observe that manifolds with divergence free Weyl
tensors and $\lfloor \frac{n-1}{2} \rfloor$-nonnegative curvature operators are
locally symmetric or conformally equivalent to a quotient of the sphere. The
main focus of the paper is to prove similar results for manifolds with special
holonomy. In particular, we consider K\"ahler manifolds with divergence free
Bochner tensor. For quaternion K\"ahler manifolds we obtain a partial result
towards the LeBrun-Salamon conjecture.
