For three high molecular weight samples of poly(dimethylsiloxane), polybutadiene, and polyisoprene the self-diffusion for times smaller than the reptation time T(rep) was investigated using field gradient NMR in the stray field of a cryomagnet. A time-dependent apparent self-diffusion coefficient D(app) is observed with time exponents b of the power law D(app) approximately t(b) between -0.57 and -0.73. These values are at variance with the strict reptation picture and support more recent theoretical concepts and computer simulations.