  Gravitational-wave observations of compact binaries have the potential to
uncover the distribution of masses and angular momenta of black holes and
neutron stars in the universe. The binary components' physical parameters can
be inferred from their effect on the phasing of the gravitational-wave signal,
but a partial degeneracy between the components' mass ratio and their angular
momenta limits our ability to measure the individual component masses. At the
typical signal amplitudes expected by the Advanced Laser Interferometer
Gravitational-wave Observatory (signal-to-noise ratios between 10 and 20), we
show that it will in many cases be difficult to distinguish whether the
components are neutron stars or black holes. We identify when the masses of the
binary components could be unambiguously measured outside the range of current
observations: a system with a chirp mass $\mathcal{M} \le 0.871 $ M$_\odot$
would unambiguously contain the smallest-mass neutron star observed, and a
system with $\mathcal{M} \ge 2.786 \Msun$ must contain a black hole. However,
additional information would be needed to distinguish between a binary
containing two 1.35 M$_\odot$ neutron stars and an exotic
neutron-star--black-hole binary. We also identify those configurations that
could be unambiguously identified as black-hole binaries, and show how the
observation of an electromagnetic counterpart to a neutron-star--black-hole
binary could be used to constrain the black-hole spin.
