We examine the cloud condensation nucleation behavior expected for dry submicron particles composed of organic species having limited solubility in water and thus exhibiting deliquescence only under water-supersaturated conditions. If saturated solution water activities aw are approximately 0.97 < aw < 1, then deliquescence is likely to control the supersaturation required to activate submicron particles, leading to high sensitivity of critical supersaturation to dry diameter. Similar behavior may occur for species, including polymeric compounds, that exhibit limited miscibility in water until very dilute conditions. Treating the sparingly-soluble or limited-miscibility species as insoluble aerosol components in calculations of their hygroscopic growth does not reproduce the predicted strong dependence of critical supersaturation on dry particle size. Lack of information on the solubility characteristics of organic-species-dominated atmospheric particulate matter may thus lead to discrepancies between predicted and measured droplet activation, even when particle composition and size are well-constrained by measurements.