A study of spectra emitted by the quiet solar corona indicates that the majority of line intensities originating in low-lying levels are consistent with isothermal plasma of ~1.3 × 106 K. Nevertheless, a number of line intensities and, in particular, those belonging to ions that are typical of higher temperatures are brighter than expected. We show in this paper that the excess brightness of the hotter lines may be satisfactorily accounted for by a two-Maxwellian electron distribution function. We have calculated the effects on the line intensities and ionization balance under the assumption of both single- and two-Maxwellian electron distribution functions. One Maxwellian is characterized by a temperature of about 110 eV (1.35 × 106 K). The second Maxwellian is assumed to be a high-energy component ranging in temperatures between 150 and 1000 eV, with electron fractions relative to the total electron density that vary from 0.5% to 10%. We found that a good match to the quiet-Sun intensities could be achieved by adding ~5% electrons with a 300-400 eV Maxwellian temperature to the cooler component at 110 eV. We also found that the calculated line intensities become inconsistent with the quiet solar corona measurements if more than 3% of a Te = 500 eV plasma or more than 1% of a Te = 1000 eV plasma is added to the cooler Maxwellian.