The irreversible strain limit epsilon(irrev) for the onset of permanent axial strain damage to Ag-sheathed Bi2Sr2Ca1Cu2O8+x and Bi2Sr2Ca2Cu3010+x superconductors has been measured to be in the range of 0.2%-0.35%. This strain damage onset is about an order of magnitude higher than for bulk sintered Y-, Bi-, or Tl-based superconductors and is approaching practical values for magnet design. The measurements show that the value of epsilon(irrev) is not dependent on magnetic field, nor does the critical current depend on strain below epsilon(irrev) at least up to 25 T at 4.2 K. Both of these factors indicate that the observed strain effect in Ag-sheathed Bi-based superconductors is not intrinsic to the superconductor material. Rather, the effect is extrinsic and arises from superconductor fracture. Thus, the damage onset is amenable to further enhancement. Indeed, the data suggest that subdividing the superconductor into fine filaments or adding Ag to the superconductor powder prior to processing significantly enhances the damage threshold epsilon(irrev) to above 0.6%.