Diffusion coefficients D of CO2 and CH4 and lifetimes of orthopositronium (o-Ps) tau(3) were measured at temperatures from 308 to 473K for polyimides, a polybenzoxazole, and poly(2,6-dimethyl-1,4-phenylene oxide) (PPO) with high glass transition temperatures T-g's above 488 K. The plots of log-(D/T) and tau(3) for these polymers at 308 K were below the correlation line observed for polymers with relatively low T-g's at: temperatures from T-g - 90 K to T-g + 70 K and interpreted by the free volume theory for diffusion. The lower D values for these high-T-g polymers were attributed to the restricted local motion of their polymer chains. With increasing temperature, the plots for PPO and a polyimide became close to the correlation line.