We find a strong correlation between the 6 cm radio and near-infrared [Fe II] 1.64 mum line emission from the nuclei of active galaxies, including starburst, LINER, and Seyfert types. For starburst galaxies this correlation is simply explained in terms of fast shocks associated with supernova remnants. There is no obvious trend with nuclear activity type, suggesting that a similar mechanism, namely shocks, is responsible for the [Fe II] and radio emission in both starburst and Seyfert galaxies, although the origin of the shocks may be different. The nuclear radio emission from all the galaxies in our sample is largely nonthermal. The Brgamma and H-2 1-0 S(1) line luminosities also correlate with the 6 cm radio emission, but the dispersion is significantly greater than found for the [Fe II] emission, indicating a secondary relationship.