A model for non-Fickian gradient diffusion in polymer-penetrant systems is studied to assess its capability in predicting experimental results in weakly non-linear differential vapor sorption experiments at conditions when "two-stage" sorption kinetics occur. The effects of six dimensionless parameters in the model on predictions of sorption kinetics are evaluated systematically by numerical simulations. The model predicts a number of observable trends. The predictions are compared successfully with sorption data of acetone vapor in VECTRA(R), a glassy random copolyester of p-hydroxybenzoic acid and 6-hydroxy-2-naphthoic acid (HBA/HNA). The results are relevant to the application of VECTRA(R) and related polymers as barrier materials.