We investigate the mechanical behavior of particle-stabilized droplets using micropipette aspiration. We observe that droplets stabilized with amphiphilic dumbbell-shaped particles exhibit a two-stage response to increasing suction pressure. Droplets first drip, then wrinkle and buckle like an elastic shell. While particles have a dramatic impact on the mechanism of failure, the mechanical strength of the droplets is only modestly increased. On the other hand, droplets coated with the molecular surfactant sodium dodecyl sulfate are even weaker than bare droplets. In all cases, the magnitude of the critical pressure for the onset of instabilities is set by the fluid surface tension.