We report a combined theoretical and experimental study of the structural failure of viral shells under mechanical stress. We find that discontinuities in the force-indentation curve associated with failure should appear when the so-called Foppl-von Karman (FvK) number exceeds a critical value. A nanoindentation study of a viral shell subject to a soft-mode instability, where the stiffness of the shell decreases with increasing pH, confirms the predicted onset of failure as a function of the FvK number.