From observations of the spectra of 15 cool metallic-line white dwarfs we derive their radial velocities from the positions of the H and K lines of Ca II. A pressure shift correction is included, and the resulting mean gravitational redshift is 35 km/s. By projecting their spatial distribution and space velocity components onto and perpendicular to the galactic plane, and including 13 additional stars not observed by us, we compare the present and past spatial distribution of the white dwarfs with the distribution of local interstellar matter. If the metallic-line white dwarfs are accreting their metals from the interstellar medium, one would expect a spatial correlation between the stars and the local gas. However, we conclude that only very few of the stars could possibly just have accreted or presently be accreting metals from local clouds