The excitation spectrum of a model magnetic system, LiHoF4, was studied with the use of neutron spectroscopy as the system was tuned to its quantum critical point by an applied magnetic field. The electronic mode softening expected for a quantum phase transition was forestalled by hyperfine coupling to the nuclear spins. We found that interactions with the nuclear spin bath controlled the length scale over which the excitations could be entangled. This generic result places a limit on our ability to observe intrinsic electronic quantum criticality.