Data from four Fermi-detected gamma-ray bursts (GRBs) are used to set limits on spectral dispersion of electromagnetic radiation across the Universe. The analysis focuses on photons recorded above 1 GeV for Fermi-detected GRB 080916C, GRB 090510A, GRB 090902B, and GRB 090926A because these high-energy photons yield the tightest bounds on light dispersion. It is shown that significant photon bunches in GRB 090510A, possibly classic GRB pulses, are remarkably brief, an order of magnitude shorter in duration than any previously claimed temporal feature in this energy range. Although conceivably a>3σ fluctuation, when taken at face value, these pulses lead to an order of magnitude tightening of prior limits on photon dispersion. Bound of Δc/c<6.94×10(-21) is thus obtained. Given generic dispersion relations where the time delay is proportional to the photon energy to the first or second power, the most stringent limits on the dispersion strengths were k1<1.61×10(-5) sec Gpc(-1) GeV(-1) and k2<3.57×10(-7) sec Gpc(-1) GeV(-2), respectively. Such limits constrain dispersive effects created, for example, by the spacetime foam of quantum gravity. In the context of quantum gravity, our bounds set M1c(2) greater than 525 times the Planck mass, suggesting that spacetime is smooth at energies near and slightly above the Planck mass.