Room temperature ionic liquid 2,3-dimethyl-1-hexylimidazolium bis(trifluoromethane sulfonyl)imide (DMHxImTFSI) has been synthesized and used in the preparation of polymer gel electrolytes containing polymethylmethacrylate and propylene carbonate (PC). The onset of ion diffusional motion has been studied by H-1 and F-19 NMR spectroscopy and the results obtained for ionic liquid, liquid electrolytes, and polymer gel electrolytes have been correlated with the ionic conductivity results for these electrolytes in the 100-400 K temperature range. The temperature at which H-1 and F-19 NMR lines show motional narrowing and hence ion diffusional motion starts has been found to be closely related to the temperature at which a large increase in ionic conductivity has been observed for these electrolytes. Polymer gel electrolytes have high ionic conductivity over a wide range of temperatures. Thermogravimetric analysis/differential scanning calorimetry studies show that the ionic liquid (DMHxImTFSI) used in the present study is thermally stable up to 400 degrees C, whereas the addition of PC lowers the thermal stability of polymer gel electrolytes containing the ionic liquid. Different electrolytes have been observed to show high ionic conductivity in different range of temperatures, which can be helpful in the design of polymer gel electrolytes for specific applications.