  It is widely believed that the spin of black holes in X-ray binaries is
mainly natal. A significant spin-up from accretion is not possible. If the
secondary has a low mass, the black hole spin cannot change too much even if
the black hole swallows the whole stellar companion. If the secondary has a
high mass, its lifetime is too short to transfer the necessary amount of matter
and spin the black hole up. However, while black holes formed from the collapse
of a massive star with Solar metallicity are expected to have low birth spin,
current spin measurements show that some black holes in X-ray binaries are
rotating very rapidly. Here I show that, if these objects are not the Kerr
black holes of general relativity, the accretion of a small amount of matter
($\sim 2$~$M_\odot$) can make them look like very fast-rotating Kerr black
holes. Such a possibility is not in contradiction with any observation and it
can explain current spin measurements in a very simple way.
