  The spin of a number of black holes (BHs) in binary systems has been
measured. In the case of BHs found in low-mass X-ray binaries (LMXBs) the
observed values are in agreement with some theoretical predictions based on
binary stellar evolution. However, using the same evolutionary models, the
calculated spins of BHs in high-mass X-ray binaries (HMXBs) fall short compared
to the observations. A possible solution to this conundrum is the accretion of
high-specific-angular-momentum material after the formation of the BH, although
this requires accretion above the Eddington limit. Another suggestion is that
the observed high values of the BHs spin could be the result of an asymmetry
during Core Collapse (CC). The only available energy to spin up the compact
object during CC is its binding energy. A way to convert it to rotational
kinetic energy is by using a Standing Accretion Shock Instability (SASI), which
can develop during CC and push angular momentum into the central compact object
through a spiral mode ($m = 1$). Here we study the CC-SASI scenario and
discuss, in the case of LMXBs and HMXBs, the limits for the spin of a
stellar-mass BHs. Our results predict a strong dichotomy in the maximum spin of
low-mass compact objects and massive BHs found in HMXBs. The maximum spin value
($|a_\star|$) for a compact object near the mass boundary between BHs and NSs
is found to be somewhere between 0.27 and 0.38, depending on whether secular or
dynamical instabilities limit the efficiency of the spin up process. For more
massive BHs, such as those found in HMXBs, the natal spin is substantially
smaller and for $M_{\rm BH}\!>\!8~M_\odot$ spin is limited to values
$|a_\star|\!\lesssim\!0.05$. Therefore we conclude that the observed high spins
of BHs in HMXBs cannot be the result of a CC-SASI spin up.
