We present lithium observations of 35 stars previously reported by Wichmann et al. (1996) to be possible new weak T Tauri stars (WTTS) discovered by ROSAT in the Taurus-Auriga star-forming region. These stars were identified on the basis of low-resolution optical spectra. We have used our higher resolution spectra for measuring the equivalent widths of the Li i 670.8 nm resonance line, and for revisiting the evolutionary status of these stars. Most ( ~ 85%) of the stars in our sample coming from ROSAT pointed observations are indeed confirmed to be new WTTS, but only a minority ( ~ 22%) of the stars coming from the ROSAT all-sky survey are confirmed as WTTS. There are two reasons why we reject some stars as WTTS. One is that seven of the stars do not have a detectable lithium line at all. The other is that we use a definition different from that Wichmann et al. (1996) for classifying stars as WTTS. In particular, we identify eight stars as post T Tauri stars (PTTS) on the basis of their moderate lithium depletion. Our results confirm that the widely dispersed RASS-selected candidate WTTS tend to be older than the T Tauri stars associated with dark molecular clouds. The presence of PTTS around central Taurus suggests that the clouds may have been forming stars for more than ~ 10 Myr, although at a very low rate. On the basis of the PTTS identified in this work we discuss possible differences between them and the WTTS. We find that PTTS seem to have slightly lower Hα emission equivalent width than WTTS, but the small number of known PTTS prevent us from making a strong conclusion. Based on observations made with the Isaac Newton and the William Herschel telescopes operated on the island of La~Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof\'i sica de Canarias