We report measurements of the electron temperature at about a dozen locations in the extended emission-line regions of five active (Seyfert and radio) galaxies. Temperatures (T_{[O III]} and T_{[N II]}) have been determined from both the I([O III] λ4363)/I([O III] λ5007) and I([N II] λ5755)/I([N II] λ6583) ratios. T_{[O III]} lies in the range (1.0-1.7) × 10⁴ K. We find a strong trend for T_{[O III]} to be higher than T_{[N II]}, with the difference typically being ≈ 5000 K. Because the critical density for collisional de-excitation of the ¹D₂ level in N II is lower than that of the same level in O III, the deviations of the measured intensity ratios from those expected for T_{[O III]} = T_{[N II]} in the low-density limit are unlikely to result from collisional de-excitation. The measured values of T_{[O III]} and the differences between T_{[O III]} and T_{[N II]} are very similar to those found in Galactic planetary nebulae. It is argued that the dominant form of energy input to the clouds is photoionization, but detailed modeling indicates that the temperature difference is too large to be accounted for in terms of photoionization of ionization-bounded clouds. We propose instead that both matter- and ionization-bounded clouds are present in the extended emission-line regions, with most of the [O III] emission originating from a hot zone in the matter-bounded clouds and essentially all the [N II] from the ionization-bounded clouds.