We have carried out an optical follow-up of twelve ROSAT discovered candidate members of the Coma Berenices open cluster. VRI photometry and low resolution ( ~ 3 Å) spectroscopy in the range 3800-7000 Åwere performed to obtain colour-magnitude diagrams, proper motions, spectral types, chromospheric activity levels and radial velocities that allow us to establish whether or not these candidates are reliable cluster members. Only four of these objects show optical photometry marginally compatible with the main sequence delineated by known cluster stars. They also exhibit spectral types corresponding to late-K and early-M, and radial velocities which are not inconsistent with membership in Coma Berenices. The proper motions of these candidates are, however, very high and incompatible with the small value associated with the cluster. Thus, none of the ROSAT candidates can be considered members of Coma Berenices. This result tends to favour the hypothesis that the mass of the cluster and its average mass density are lower than what is required for stability, allowing the cluster dissolution by the escape of the less massive stars. We discuss the probable nature of these objects on the basis of the available information. From our radial velocity observations we infer that five of them could be formed by multiple stars. In particular, we have discovered one W-type W UMa contact binary system (K6 spectral type) which is, in addition, a visual companion to a cooler star. Based on observations made with the Isaac Newton telescope, operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos, with the IAC80 telescope and with the European Space Agency OGS telescope operated on the island of Tenerife by the Instituto de Astrofísica de Canarias in the Spanish Observatorio del Teide of the Instituto de Astrofísica de Canarias.