$K^-$ mesons offer a unique setting where mesic atoms have been
studied both experimentally and theoretically, thereby placing
constraints on the possible existence and properties of
meson-nuclear quasibound states. Here we review progress
in this field made recently by the Jerusalem--Prague
Collaboration using near-threshold $K^-N$ scattering amplitudes
generated in several meson--baryon coupled channels models
inspired by a chiral EFT approach.
Our own procedure of handling subthreshold kinematics self
consistently is used to transform these free-space energy
dependent amplitudes to in-medium density dependent amplitudes
from which $K^-$ optical potentials are derived.
To fit the world data of kaonic atoms, these single-nucleon
optical potentials are augmented by multi-nucleon terms.
It is found that only two of the studied models reproduce
also the single-nucleon absorption fractions available from
old bubble chamber experiments. These two models are
then checked for possible $K^-$ nuclear quasibound states,
despite realizing that $K^-$ optical potentials are not
constrained by kaonic atom data at densities exceeding
half nuclear-matter density. We find that when such states
exist, their widths are invariably above 100 MeV,
forbiddingly large to allow observation. Multi-nucleon
absorption is found to be substantial in this respect.
This suggests that observable strongly bound $K^-$ mesons
are limited to the very light systems, such as $K^-pp$.