  Bimetric gravity theories allow for many different types of cosmological
solutions, but not all of them are theoretically allowed. In this work we
discuss the conditions to satisfy the Higuchi bound and to avoid gradient
instabilities in the scalar sector at the linear level. We find that in
expanding universes the ratio of the scale factors of the reference and
observable metric has to increase at all times. This automatically implies a
ghost-free helicity-2 and helicity-0 sector and enforces a phantom dark energy.
Furthermore, the condition for the absence of gradient instabilities in the
scalar sector will be analyzed. Finally, we discuss whether cosmological
solutions can exist, including exotic evolutions like bouncing cosmologies, in
which both the Higuchi ghost and scalar instabilities are absent at all times.
