Due to the difficulty of performing uniaxial tensile testing, the strengths of graphene and its grain boundaries have been measured in experiments by nanoindentation testing. From a series of molecular dynamics simulations, we find that the strength measured in uniaxial simulation and the strength estimated from the nanoindentation fracture force can differ significantly. Fracture in tensile loading occurs simultaneously with the onset of crack nucleation near 5-7 defects, while the graphene sheets often sustain the indentation loads after the crack initiation because the sharply concentrated stress near the tip does not give rise to enough driving force for further crack propagation. Due to the concentrated stress, strength estimation is sensitive to the indenter tip position along the grain boundaries. Also, it approaches the strength of pristine graphene if the tip is located slightly away from the grain boundary line. Our findings reveal the limitations of nanoindentation testing in quantifying the strength of graphene, and show that the loading-mode-specific failure mechanism must be taken into account in designing reliable devices from graphene and other technologically important 2D materials.