We study numerically the failure of an interface joining two elastic materials under load using a fiber bundle model connected to an elastic half space. We find that the breakdown process follows the equal load sharing fiber bundle model without any detectable spatial correlations between the positions of the failing fibers until localization sets in. Depending on the elastic constant describing the elastic half space, localization sets in before or after the critical load causing the interface to fail completely is reached. There is a crossover between failure due to localization and failure without spatial correlations when tuning the elastic constant, not a phase transition as has been proposed earlier. Also, contrary to earlier claims based on models different from ours, we find that a finite fraction of fibers must fail before the critical load is attained, even in the extreme localization regime, i.e., for very small elastic constant. We furthermore find that the critical load remains finite for all values of the elastic constant in the limit of an infinitely large system.