Fluid injection stimulates seismicity far from active tectonic regions, however the details of how fluids modify on-fault stresses and initiate seismic events remains poorly understood. We conducted laboratory experiments using a biaxial loading apparatus with a 3 m saw-cut granite fault and compared events induced at different background shear stress levels. Water was injected at 10 mL/min and normal stress was constant at 4 MPa. In all experiments, aseismic slip initiated on the fault near the location of fluid injection and dynamic rupture eventually initiated from within the aseismic slipping patch. When the fault was near critically stressed, seismic slip initiated only seconds after MPa-level injection pressures were reached and the dynamic rupture propagated beyond the fluid pressure perturbed region. At lower stress levels, dynamic rupture initiated hundreds of seconds later and was limited to regions where aseismic slip had significantly redistributed stress from within the pressurized region to neighboring locked patches. We find that slow slip initiated when local stresses exceeded Coulomb failure criteria, but initiation of dynamic rupture required additional criteria to be met. Even high background stress levels required aseismic slip to modify on-fault stress to meet initiation criteria. We also observed slow slip events prior to dynamic rupture. Overall, our experiments suggest that initial fault stress, relative to fault strength, is a critical factor in determining whether a fluid-induced rupture will “runaway” or whether a fluid-induced rupture will remain localized to the fluid pressurized region.