The progenitors of many Type II core-collapse supernovae (SNe) have now been identified directly on pre-discovery imaging. Here, we present an extensive search for the progenitors of Type Ibc SNe in all available pre-discovery imaging since 1998. There are 12 Type Ibc SNe with no detections of progenitors in either deep ground-based or Hubble Space Telescope archival imaging. The deepest absolute BVR magnitude limits are between -4 and -5mag. We compare these limits with the observed Wolf-Rayet population in the Large Magellanic Cloud and estimate a 16 percent probability that we have failed to detect such a progenitor by chance. Alternatively, the progenitors evolve significantly before core-collapse or we have underestimated the extinction towards the progenitors. Reviewing the relative rates and ejecta mass estimates from light-curve modelling of Ibc SNe, we find both incompatible with Wolf-Rayet stars with initial masses >25M_{sun}_ being the only progenitors. We present binary evolution models that fit these observational constraints. Stars in binaries with initial masses <~20M_{sun}_ lose their hydrogen envelopes in binary interactions to become low-mass helium stars. They retain a low-mass hydrogen envelope until ~10^4^yr before core-collapse; hence, it is not surprising that Galactic analogues have been difficult to identify.