While the notion that injecting fluids into the subsurface can reactivate faults by reducing frictional resistance is well established, the ensuing evolution of slip is still poorly understood. What controls whether the induced slip remains stable and confined to the fluid-affected zone or accelerates into a runaway earthquake? Are there observable indicators of the propensity to earthquakes before they happen? Here, we investigate these questions by modeling a unique fluid-injection experiment on a natural fault with laboratory-derived friction laws. We show that a range of fault models with diverging stability with sustained injection reproduce the slip measured during pressurization. Upon depressurization, however, the most unstable scenario departs from the observations, suggesting that the fault is relatively stable. The models could be further distinguished with optimized depressurization tests or spatially distributed monitoring. Our findings indicate that avoiding injection near low-residual-friction faults and depressurizing upon slip acceleration could help prevent large-scale earthquakes.