  The excitation spectrum of a model magnetic system, LiHoF$_4$, has been
studied using neutron spectroscopy as the system is tuned to its quantum
critical point by an applied magnetic field. The electronic mode softening
expected for a quantum phase transition is forestalled by hyperfine coupling to
the nuclear spins. We show that interactions with the nuclear spin bath control
the length scale over which the excitations can be entangled. This generic
result limits how far it is possible to approach intrinsic electronic quantum
criticality.
