  Pressure is a unique tuning parameter for probing the properties of materials
and has been particularly useful for studies of electronic materials such as
high-temperature cuprate superconductors. Here we report the effects of
quasi-hydrostatic compression produced by a neon pressure-medium on the
structures of bismuth-based high $\mathit{T_c}$ cuprate superconductors with
the nominal composition $Bi_2Sr_2Ca_{n-1}Cu_nO_{2n+4+\delta}$ (n=1,2,3) up to
155 GPa. The structures of all three compositions obtained by synchrotron X-ray
diffraction can be described as pseudo-tetragonal over the entire pressure
range studied. We show that previously reported pressure-induced distortions
and structural changes arise from the large strains that can be induced in
these layered materials by non-hydrostatic stresses. The pressure-volume
equations of state (EOS) measured under these quasi-hydrostatic conditions
cannot be fit to single phenomenological formulation over the pressure ranges
studied, starting below 20 GPa. This intrinsic anomalous compression as well as
the sensitivity of $Bi_2Sr_2Ca_{n-1}Cu_nO_{2n+4+\delta}$ to deviatoric stresses
provides explanations for the numerous inconsistencies in reported EOS
parameters for these materials. We conclude that the anomalous compressional
behavior of all three compositions is a manifestation of the changes in
electronic properties that are also responsible for the remarkable
non-monotonic dependence of $\mathit{T_c}$ with pressure, including the
increase in $\mathit{T_c}$ at the highest pressures studied so far for each.
Transport and spectroscopic measurements up to megabar pressures are needed to
fully characterize and explore still higher possible critical temperatures in
these materials.
