A simple test for time asymmetry is devised and carried out on the brightest gamma-ray bursts (GRBs) detected by the Burst and Transient Source Experiment (BATSE) on board the Compton Gamma Ray Observatory. We show evidence that individual bursts are time-asymmetric on all timescales tested, from a timescale shorter than that of pulses which compose GRBs to a timescale similar to a greater envelope that contains these pulses. We also find bursts which manifest significant asymmetry only on timescales comparable to the duration of burst, and bursts for which no clear asymmetry on any timescale is present. The sense of the asymmetry is that bursts and/or component structures rise in a shorter time than they decay. We also find that our whole sample of bursts taken together is time-asymmetric, in that there are sifnificantly more bursts and pulses where the rise is more rapid than the decay, on all timescales tested and for all energy bands tested. When our whole GRB sample is binned at 64 ms and integrated over all BATSE energies, the statistical significance is at the 6 sigma level. Models that predict time symmetry are therefore excluded.