  In this article we study the concept of quaternionic bisectional curvature
introduced by B. Chow and D. Yang for quaternion-K\"ahler manifolds. We show
that non-negative quaternionic bisectional curvature is only realized for the
quaternionic projective space. We also show that all symmetric
quaternion-K\"ahler manifolds different from the quaternionic projective space
admit quaternionic lines of negative quaternionic bisectional curvature. In
particular this implies that non-negative sectional curvature does not imply
non-negative quaternionic bisectional curvature. Moreover we give a new and
rather short proof of a classification result by A. Gray on compact K\"ahler
manifolds of non-negative sectional curvature.
