We present high-resolution, high-cadence observations of six, fine-scale, on-disk jet-like events observed by the High-resolution Coronal Imager 2.1 (Hi-C 2.1) during its sounding-rocket flight. We combine the Hi-C 2.1 images with images from the Solar Dynamics Observatory (SDO)/Atmospheric Imaging Assembly (AIA) and the Interface Region Imaging Spectrograph (IRIS), and investigate each event's magnetic setting with co-aligned line-of-sight magnetograms from the SDO/Helioseismic and Magnetic Imager (HMI). We find that (i) all six events are jetlet-like (having apparent properties of jetlets), (ii) all six are rooted at edges of magnetic network lanes, (iii) four of the jetlet-like events stem from sites of flux cancelation between majority-polarity network flux and merging minority-polarity flux, and (iv) four of the jetlet-like events show brightenings at their bases reminiscent of the base brightenings in coronal jets. The average spire length of the six jetlet-like events (9000 ± 3000 km) is three times shorter than that for IRIS jetlets (27,000 ± 8000 km). While not ruling out other generation mechanisms, the observations suggest that at least four of these events may be miniature versions of both larger-scale coronal jets that are driven by minifilament eruptions and still-larger-scale solar eruptions that are driven by filament eruptions. Therefore, we propose that our Hi-C events are driven by the eruption of a tiny sheared-field flux rope, and that the flux rope field is built and triggered to erupt by flux cancelation.