  Bose-Einstein condensates (BECs) have been proposed as candidate states of
matter for the interior of neutron stars. Specifically, Chavanis and Harko
obtained the mass-radius relation for a BEC star and proposed that the recently
discovered neutron stars with masses around 2$M_\odot$ are BEC stars. They
employed a barotropic equation of state (EOS), with one free parameter, that
was first found by Colpi, Wasserman, and Shapiro (CSW), to describe them and
derive stable equilibrium configurations of spinning BEC stars in General
Relativity. In this work we show that while it is true that BECs allow for
compact object masses as heavy as the heaviest observed ones, such stars cannot
simultaneously have radii that are small enough to be consistent with the
latest observations, in spite of the flexibility available in the EOS in the
form of the free parameter. In fact, our conclusion applies to any spinning
relativistic boson star that obeys the CSW EOS.
