We present mid-resolution spectroscopic observations of 35 stars reported to be weak T Tauri stars (WTTS) discovered by ROSAT in the Taurus-Auriga star-forming region (Wichmann et al. 1996). We have measured the equivalent widths of Hα and Li 1 670.8 nm in our spectra. On the basis of our higher-resolution data, we reassess the evolutionary status of these stars. Most of the stars in the Wichmann sample coming from ROSAT pointed observations are confirmed to be WTTSs. However, only a minority (18%) of the stars discovered in the ROSAT all-sky survey (RASS) really are WTTSs. It appears that the RASS sources are mainly older stars (PTTSs, ZAMS and dMe). Our results do not support previous suggestions about the existence of a numerous population of WTTSs widely dispersed around molecular clouds.