When grid-forming (GFM) inverter-based resources (IBRs) face severe grid disturbances (e.g., short-circuit faults), the current limitation mechanism may be triggered. Consequently, the GFM IBRs enter the current-saturation mode, inducing nonlinear dynamical behaviors and posing great challenges to the post-disturbance transient angle stability. This paper presents a systematic study to reveal the fault recovery behaviors of a GFM IBR and identify the risk of instability. A closed-form expression for the necessary condition that a GFM IBR returns from the current-saturation mode to the normal operation mode is presented. Based on these analyses, it is inferred that the angle of the magnitude-saturated current significantly affects the post-fault recovery and transient stability; with different angle selection, the system may follow multiple post-fault trajectories depending on those conditions: 1) Convergence to a normal stable equilibrium point (SEP), 2) convergence to a saturated stable equilibrium point (satSEP), or 3) divergence (instability). In this paper, the circumstances under which a GFM IBR cannot escape from the current-saturation mode are thoroughly investigated. The theoretical analyses are verified by dynamic simulations.