Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X ⊂ Y , we study the question of when a bundle E on X, extends to a bundle e on a Zariski open set U ⊂ Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck-Lefschetz theory. As a consequence, we prove a Noether-Lefschetz theorem for higher rank bundles, which recovers and unifies the NoetherLefschetz theorems of Joshi and Ravindra-Srinivas.