  The single ion physics of Ho$_2$Ti$_2$O$_7$ is well-understood to produce
strong Ising anisotropy, which is an essential ingredient to its
low-temperature spin ice state. We present inelastic neutron scattering
measurements on Ho$_2$Ti$_2$O$_7$ that reveal a clear inconsistency with its
established single ion Hamiltonian. Specifically, we show that a crystal field
doublet near 60~meV is split by approximately 3~meV. Furthermore, this crystal
field splitting is not isolated to Ho$_2$Ti$_2$O$_7$ but can also be found in
its chemical pressure analogs, Ho$_2$Ge$_2$O$_7$ and Ho$_2$Sn$_2$O$_7$. We
demonstrate that the origin of this effect is a vibronic bound state, resulting
from the entanglement of a phonon and crystal field excitation. We derive the
microscopic Hamiltonian that describes the magneto-elastic coupling and
provides a quantitative description of the inelastic neutron spectra.
