We carry out plasma diagnostic analyses for 123 planetary nebulae (PNe) and 42 HII regions using the NII and OII optical recombination lines (ORLs). New effective recombination coefficients for the NII and OII optical recombination spectra are used. These data were calculated under the intermediate coupling scheme for a number of electron temperature (Te) and density (Ne) cases. We use a new method to determine the Te's and Ne's for the nebular sample, combining the ORLs with the most reliable measurements for each ion and the predicted intensities that are based on the new atomic data. Uncertainties of the derived Te and Ne are estimated for each object. The diagnostic results from heavy element ORLs show reasonable agreement with previous calculations in the literature. We compare the electron temperatures derived from the NII and OII ORLs, Te(ORLs), and those from the collisionally excited lines (CELs), Te(CELs), as well as the hydrogen Balmer jump, Te(HIBJ), especially for the PNe with large abundance discrepancies. Temperatures from the HeI recombination lines, Te(HeI), are also used for comparison if available. For all the objects included in our sample, Te(ORLs) are lower than Te(HIBJ), which are in turn systematically lower than Te(CELs). PNe with Te(HeI) available show the relation Te(ORLs)<=Te(HeI)<=Te(HIBJ)<=Te(CELs), which is consistent with predictions from the bi-abundance nebular model postulated by Liu et al. (2000MNRAS.312..585L).