We measure the upper and lower limits of the temperatures Ti for a number of ions observed in the quiescent solar corona from 1996 to 2000, in order to compare with the measured electron temperature Te and test the assumption Ti=Te commonly adopted to measure the plasma nonthermal velocity vnth. We use a diagnostic technique that relaxes all assumptions on Ti and only requires that the plasma nonthermal velocity be the same for all ions; this provides a measurement of the upper limit on vnth and of a range of Ti-values compatible with observations. The measurements were carried out using spectra observed outside the solar limb in 1996, 1999, and 2000, spanning the rising phase of the solar cycle, and are repeated at different heights above the solar photosphere for each data set. We find that in the 2000 data set Ti>Te in the majority of cases, while the differences are smaller in the 1999 data set and nearly absent in the 1996 data set. We also investigate the correlation of the measured values of Ti with line wavelength, ion charge, charge-to-mass ratio, and temperature of maximum abundance, line intensity and width, and atomic weight, finding none.