We study the buckling of hemispherical elastic shells subjected to the combined effect of pressure loading and a probing force. We perform an experimental investigation using thin shells of nearly uniform thickness that are fabricated with a well-controlled geometric imperfection. By systematically varying the indentation displacement and the geometry of the probe, we study the effect that the probe-induced deflections have on the buckling strength of our spherical shells. The experimental results are then compared to finite element simulations, as well as to recent theoretical predictions from the literature. Inspired by a nondestructive technique that was recently proposed to evaluate the stability of elastic shells, we characterize the nonlinear load-deflection mechanical response of the probe for different values of the pressure loading. We demonstrate that this nondestructive method is a successful local way to assess the stability of spherical shells.