  We prove that a compact Riemannian manifold of dimension $m \geq 3$ with
harmonic curvature and $\lfloor\frac{m-1}{2}\rfloor$-positive curvature
operator has constant sectional curvature, extending the classical Tachibana
theorem for manifolds with positive curvature operator. The condition of
$\lfloor\frac{m-1}{2}\rfloor$-positivity originates from recent work of
Petersen and Wink, who proved a similar Tachibana-type theorem under the
stronger condition that the manifold be Einstein. We show that the same
rigidity property holds for complete manifolds assuming either parabolicity, an
integral bound on the Weyl tensor or a stronger pointwise positive lower bound
on the average of the first $\lfloor\frac{m-1}{2}\rfloor$ eigenvalues of the
curvature operator. For 3-manifolds, we show that positivity of the curvature
operator can be relaxed to positivity of the Ricci tensor.
