Energetic feedback from active galactic nuclei (AGN) is an important ingredient for regulating the star-formation history of galaxies in models of galaxy formation, which makes it important to study how AGN feedback actually occurs in practice. In order to catch AGNs in the act of quenching star formation we have used the interstellar NaD absorption lines to look for cold-gas outflows in a sample of 456 nearby galaxies for which we could unambigously ascertain the presence of radio AGN activity, thanks to radio imaging at milli-arcsecond scales. While compact radio emission indicating a radio AGN was found in 103 galaxies (23% of the sample), and 23 objects (5%) exhibited NaD absorption-line kinematics suggestive of cold-gas outflows, not one object showed evidence of a radio AGN and of a cold-gas outflow simultaneously. Radio AGN activity was found predominantly in early-type galaxies, while cold-gas outflows were mainly seen in spiral galaxies with central star-formation or composite star-formation/AGN activity. Optical AGNs also do not seem capable of driving galactic winds in our sample. Our work adds to a picture of the low-redshift Universe where cold-gas outflows in massive galaxies are generally driven by star formation and where radio-AGN activity occurs most often in systems in which the gas reservoir has already been significantly depleted.