We present an analysis of the properties of neutral hydrogen (H i) in 248 nearby (0.02 < z < 0.25) radio galaxies with S 1.4 GHz > 30 mJy and for which optical spectroscopy is available. The observations were carried out with the Westerbork Synthesis Radio Telescope as the last large project before the upgrade of the telescope with phased array feed receivers (Apertif). The sample covers almost four orders of magnitude in radio power from log P 1.4 GHz = 22.5 W Hz −1 and 26.2 W Hz −1 . We detect H i in absorption in 27% ± 5.5% of the objects. The detections are found over the full range of radio power. However, the distribution and kinematics of the absorbing H i gas appear to depend on radio power, the properties of the radio continuum emission, and the dust content of the sources. Among the sources where H i is detected, gas with kinematics deviating from regular rotation is more likely found as the radio power increases. In the great majority of these cases, the H i profile is asymmetric with a significant blue-shifted component. This is particularly common for sources with log P 1.4 GHz > 24 W Hz −1 , where the radio emission is small, possibly because these radio sources are young. The same is found for sources that are bright in the mid-infrared, i.e. sources rich in heated dust. In these sources, the H i is outflowing likely under the effect of the interaction with the radio emission. Conversely, in dust-poor galaxies, and in sources with extended radio emission, at all radio powers we only detect H i distributed in a rotating disk. Stacking experiments show that in sources for which we do not detect H i in absorption directly, the H i has a column density that is lower than 3.5 × 10 17 (T spin /c f ) cm −2 . We use our results to predict the number and type of H i absorption lines that will be detected by the upcoming surveys of the Square Kilometre Array precursors and pathfinders (Apertif, MeerKAT, and ASKAP).