We present the results of deep Westerbork Synthesis Radio Telescope observations of neutral hydrogen in 12 nearby elliptical and lenticular galaxies. The selected objects come from a representative sample of nearby galaxies earlier studied at optical wavelengths with the integral-field spectrograph SAURON (Spectrographic Areal Unit for Research on Optical Nebulae). They are field galaxies, or (in two cases) located in poor group environments. We detect H I - both in regular discs as well as in clouds and tails offset from the host galaxy - in 70 per cent of the galaxies. This detection rate is much higher than in previous, shallower single-dish surveys, and is similar to that for the ionized gas. The results suggest that at faint detection levels the presence of H I is a relatively common characteristic of field early-type galaxies, confirming what was suggested twenty years ago by Jura based on IRAS observations. The observed total H I masses range between a few times 10(6) to just over 10(9) M(circle dot) . The presence of regular disc-like structures is a situation as common as H I in offset clouds and tails around early-type galaxies. All galaxies where H I is detected also contain ionized gas, whereas no H I is found around galaxies without ionized gas. Galaxies with regular H I discs tend to have strong emission from ionized gas. In these cases, the similar kinematics of the neutral hydrogen and ionized gas suggest that they form one structure. The kinematical axis of the stellar component is nearly always misaligned with respect to that of the gas. We do not find a clear trend between the presence of H I and the global age of the stellar population or the global dynamical characteristics of the galaxies. More specifically, H I detections are uniformly spread through the (V/sigma, epsilon) diagram. If fast and slow rotators - galaxies with high and low specific angular momentum - represent the relics of different formation paths, this does not appear in the presence and characteristics of the H I. Our observations support the idea that gas accretion is common and does not happen exclusively in peculiar early-type galaxies. The links observed between the large-scale gas and the characteristics on the nuclear scale (e.g. the presence of kinematically decoupled cores, radio continuum emission etc.) suggest that for the majority of the cases the gas is acquired through merging, but the lack of correlation with the stellar population age suggests that smooth, cold accretion could be an alternative scenario, at least in some galaxies. In either case, the data suggest that early-type galaxies continue to build their mass up to the present.