Given a compact complex manifold M , we investigate the holomorphic vector bundles E on M such that ϕ * E is trivial for some surjective holomorphic map ϕ, to M , from some compact complex manifold. We prove that these are exactly those holomorphic vector bundles that admit a flat holomorphic connection with finite monodromy homomorphism. A similar result is proved for holomorphic principal G-bundles, where G is a connected reductive complex affine algebraic group.