We study the effects that a curvature correction to the line tension has on the equilibrium and stability of liquid droplets laid upon a rigid substrate. In the simple case of cylindric liquid bridges we prove that even a tiny curvature correction prevents the onset of wildly oscillating perturbations that would make the contact line unstable if a negative line tension were present alone. However, if the curvature correction is not large enough, unstable modes that are not related to the classical Rayleigh instability can persist.