We revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell. By examining the energy landscape of the perfect cylinder, we deduce an estimate of the sensitivity of the shell to imperfections. Key to obtaining this estimate is the existence of a mountain pass point for the system. We prove the existence on bounded domains of such solutions for almost all loads and then numerically compute example mountain pass solutions. Numerically the mountain pass solution with lowest energy has the form of a single dimple. We interpret these results and validate the lower bound against some experimental results available in the literature.