Motivated by recent experimental observations [Rowley et al. in Phys Rev 96:020407, 2017] on hexagonal ferrites, we revisit the phase diagrams of diluted magnets close to the lattice percolation threshold. We perform large-scale Monte Carlo simulations of XY and Heisenberg models on both simple cubic lattices and lattices representing the crystal structure of the hexagonal ferrites. Close to the percolation threshold pc, we find that the magnetic ordering temperature Tc depends on the dilution p via the power law Tc ∼|p - pc|Φ with exponent Φ = 1.09, in agreement with classical percolation theory. However, this asymptotic critical region is very narrow, |p - pc| ≲ 0.04. Outside of it, the shape of the phase boundary is well described, over a wide range of dilutions, by a nonuniversal power law with an exponent somewhat below unity. Nonetheless, the percolation scenario does not reproduce the experimentally observed relation Tc ~ (xc -x)2/3 in PbFe12-xGaxO19. We discuss the generality of our findings as well as implications for the physics of diluted hexagonal ferrites.