When a droplet approaches a solid surface, the thin liquid film between the droplet and the surface drains until an instability forms and then ruptures. In this study, we utilize microfluidics to investigate the effects of film thickness on the time to film rupture for water droplets in a flowing continuous phase of silicone oil depositing on solid poly(dimethylsiloxane) (PDMS) surfaces. The water droplets ranged in size from millimeters to microns, resulting in estimated values of the film thickness at rupture ranging from 600 nm down to 6~nm. The Stefan-Reynolds equation is used to model film drainage beneath both millimeter- and micron-scale droplets. For millimeter-scale droplets, the experimental and analytical film rupture times agree well, while large differences are observed for micron-scale droplets. We speculate that the differences in the micron-scale data result from the increases of the local thin film viscosity due to confinement-induced molecular structure changes of the silicone oil. A modified Stefan-Reynolds equation is used to account for the increased thin film viscosity for the micron-scale droplet drainage case.