Let E → B be a fibre bundle and let Eʹ → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and Eʹ – 0, where 0 is the zero section of Eʹ → B. Let f : E → Eʹ be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 𝕊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions.