We conducted a 12-month monitoring campaign of 33 T Tauri stars (TTS) in Taurus. Our goal was to monitor objects that possess a disk but have a weak Halpha line, a common accretion tracer for young stars, to determine whether they host a passive circumstellar disk. We used medium-resolution optical spectroscopy to assess the objects' accretion status and to measure the Halpha line. We found no convincing example of passive disks; only transition disk and debris disk systems in our sample are non-accreting. Among accretors, we find no example of flickering accretion, leading to an upper limit of 2.2% on the duty cycle of accretion gaps assuming that all accreting TTS experience such events. Combining literature results with our observations, we find that the reliability of traditional Halpha-based criteria to test for accretion is high but imperfect, particularly for low-mass TTS. We find a significant correlation between stellar mass and the full width at 10 per cent of the peak (W10%) of the Halpha line that does not seem to be related to variations in free-fall velocity. Finally, our data reveal a positive correlation between the Halpha equivalent width and its W10%, indicative of a systematic modulation in the line profile whereby the high-velocity wings of the line are proportionally more enhanced than its core when the line luminosity increases. We argue that this supports the hypothesis that the mass accretion rate on the central star is correlated with the Halpha W10% through a common physical mechanism.< Réduire