We study the spectral characteristics of 93 impulsive, hard X-ray flares that were observed by the hard X-ray burst spectrometer (HXRBS) on the Solar Maximum Mission (SMM) spacecraft. Our major findings are as follows (1) During the initial few seconds after onset of rapidly rising bursts (i.e., rise times less-than-or-similar-to 5 s), there is a "high-energy delay," where the rise in flux at energies greater-than-or-similar-to 150 keV is delayed by a few seconds relative to that at energies less-than-or-similar-to 100 keV. (2) At the times of peak flux, the power-law spectra almost always "break downward" at an energy of approximately 100 keV, i.e., the spectra at higher energies are steeper than those at lower energies. There is little or no dependence of break energy upon the hard X-ray flux. (3) During the decay phase of bursts, the slopes of the power-law distributions at high and low energies change or cross over in such a way that the overall spectrum assumes the form of either a single power law or a broken power law that breaks up. The break energies of the spectra are usually lower after the crossover, but in approximately 30% of the cases they are higher. We relate our observational results to models of solar flares that invoke electric fields as a particle acceleration mechanism. In this context, our results imply that (a) the electric fields commonly have a potential drop of 150-200 keV, (b) another process (possibly stochastic acceleration by waves generated in association with the particle dynamics) produces the electrons of greater-than-or-similar-to 200 keV, and this process requires a few seconds to operate, and (c) as bursts decay the electric field disappears and the electron distribution evolves to a simple power law, or one that breaks up because of the preferential loss of electrons of E less-than-or-similar-to 100 keV, caused by collisions or by cyclotron maser radiation.