We present the results of deep Westerbork Synthesis Radio Telescope observations of the neutral hydrogen in 12 nearby elliptical and lenticular galaxies selected from a representative sample of nearby galaxies that were studied earlier at optical wavelengths with the integral-field spectrograph sauron. The observed objects are field galaxies, or (in two cases) are located in poor-group environments. We detect Hi in 70% of the galaxies. This detection rate is much higher than in previous, shallower single-dish surveys, and is similar to that for the ionised gas. The results suggest that at faint detection levels the presence of Hi is a relatively common characteristic of field early-type galaxies. The presence of regular disc-like structures is as common as Hi in offset clouds and tails. All galaxies where Hi is detected also contain ionised gas, whereas no Hi is found around galaxies without ionised gas. Galaxies with regular Hi discs tend to have strong emission from ionised gas. In these cases, the similar kinematics of the neutral hydrogen and ionised gas suggest that they form one structure. We do not find a trend between the presence of Hi and the global age of the stellar population or with the global dynamical characteristics of the galaxies. If fast and slow rotators represent the relics of different formation paths, this does not appear in the presence and characteristics of the Hi. The links observed between the large-scale gas and the characteristics on the nuclear scale (e.g. the presence of kinematically decoupled cores and radio continuum emission), 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-up their mass to the present.