Variability detected in galaxies is usually attributed to their active galactic nuclei (AGNs). While all AGNs are intrinsically variable, the AGN unification model predicts that Type 2 AGNs rarely vary because their engines are blocked by dust tori. Previous UV to near-IR variability studies largely support this expectation. Here, we present a variability study with the James Webb Space Telescope (JWST) that reveals a more subtle picture. Using NIRCam imaging data from three surveys over ∼140arcmin²in the COSMOS field, we found 117 galaxies with ≥4σvariability in the F356W band across an ∼2 yr baseline. Crossmatching with existing JWST spectroscopic data, we identified five of them atz= 0.19–3.69 (F356W corresponding to rest-frameλ≈0.76–2.97μm), which were all coincidentally observed by an NIRSpec program almost contemporaneously with the last imaging epoch. One additional variable was identified atz= 0.90 using archival Keck telescope data. These six objects form our spectroscopic subsample. Interestingly, two reside in close-pair environments, while two others form a close pair themselves. Most of their light curves can hardly be explained by nuclear transients, and AGN variability is a more plausible cause. However, among these six objects (1) only one shows broad Bracket and Pfund series permitted lines (Δv>1000kms⁻¹) indicative of a Type 1 AGN; (2) two show narrow permitted lines (Hαand/or HeIλ10830) consistent with Type 2 AGNs, with another one likely Type 2 based on the host galaxy properties; and (3) two others, which form a pair, show no emission lines. Our results add more challenges to the unification model.