Groundwater in aquifers is generally older than expected on the basis of flow velocity, and this observation has important implications for interpreting radiometric age determinations. Hydrologists commonly account for the aging of water as it flows along streamtubes, but not for the effects of mixing old water from aquitards (or confining layers) into aquifers, because the rate of mass exchange between aquifers and aquitards can in many cases be assumed to be small. We show, however, that the effect on age of such mixing does not depend on the mixing rate; this is the paradox of groundwater age. Surprisingly, the contribution of aquitards to the age of groundwater in aquifers depends only on the ratio of fluid volume in aquitards to aquifers. This result has broad importance for understanding the relationship between groundwater flow and the distribution of radiometric age.