  We present a systematic study of the phase diagram of LiHo$_x$Y$_{1-x}$F$_4$
($0.25<x<1$) Ising ferromagnets obtained from neutron scattering measurements
and mean-field calculations. We show that while the thermal phase transition
decreases linearly with dilution, as predicted by mean-field theory, the
critical transverse field at the quantum critical point is suppressed much
faster. This behavior is related to competition between off-diagonal dipolar
coupling and quantum fluctuations that are tuned by doping and applied field,
respectively. In this paper, we quantify the deviation of the experimental
results from mean-field predictions, with the aim that this analysis can be
used in future theoretical efforts towards a quantitative description.
