We report on direct time-of-flight based mass measurements of 16 light neutron-rich nuclei. These include the first determination of the masses of the Borromean drip-line nuclei (19)B, (22)C, and (29)F as well as that of (34)Na. In addition, the most precise determinations to date for (23)N and (31)Ne are reported. Coupled with recent interaction cross-section measurements, the present results support the occurrence of a two-neutron halo in (22)C, with a dominant ν2s(1/2)(2) configuration, and a single-neutron halo in (31)Ne with the valence neutron occupying predominantly the 2p(3/2) orbital. Despite a very low two-neutron separation energy the development of a halo in (19)B is hindered by the 1d(5/2)(2) character of the valence neutrons.