We consider nanometer-sized fluid annuli (rings) deposited on a solid substrate and ask whether these rings break up into droplets due to the instability of Rayleigh-Plateau-type modified by the presence of the substrate, or collapse to a central drop due to the presence of azimuthal curvature. The analysis is carried out by a combination of atomistic molecular dynamics simulations and a continuum model based on a long-wave limit of Navier-Stokes equations. We find consistent results between the two approaches, and demonstrate characteristic dimension regimes which dictate the assembly dynamics.