The speed at which an annular liquid collar drains under gravity g in a vertical tube of radius a, when the tube has an otherwise thin viscous liquid lining on its interior, is determined by a balance between the collar's weight and viscous shear stresses confined to narrow regions in the neighborhood of the collar's effective contact lines. Whether a collar grows or shrinks in volume as it drains depends on the modified Bond number B = rho ga(2)/(sigma epsilon), where rho is the fluid density, sigma is its surface tension, and epsilon a is the thickness of the thin film immediately ahead of the collar. Asymptotic methods are used here to determine the following nonlinear stability criteria for an individual collar, valid in the limit of small epsilon. For 0 < B < 0.5960, all draining collars grow in volume and, in sufficiently long tubes, ultimately "snap off'' to form stable lenses. For 0.5960 < B < 1.769, small collars may shrink but in long tubes sufficiently large collars will snap off. For 1.769 < B < 11.235, both stable collars and lenses may arise, although most collars Rill shrink. If B > 11.235, all collars and lenses shrink in volume as they drain, so that any lens ultimately ruptures, unless stabilizing intermolecular forces allow the formation of a lamella supported by a macroscopic Plateau border. If surfactant immobilizes the liquid's free surface, these critical values of B are reduced by a factor of 2 but the distance a collar must travel before it snaps off is unchanged. Gravitationally driven snap off is therefore most likely to occur in long tubes with radii substantially less than the capillary lengthscale (sigma/rho g)(1/2).