Let me answer your second question first.
In some cases there certainly is a threshold for magnetism. An example on which I've worked a little is gold with some alloyed iron: Au1-x
. At low fractions x of Fe no ordinary magnetic state is formed. Instead, at low enough temperature the magnetic moments freeze into a peculiar spinglass state. The lower x is, the colder the spinglass freezing temperature. At higher x, when the material is cold enough it forms ferromagnetism. If I remember right, that happens above x of around 0.15 or so. That's around the percolation threshold at which you start to find clusters of Fe which are connected all across the sample. Again, as you raise x the freezing temperature goes up.
The prediction of the magnetic states of different crystals is possible but difficult. The techniques used are essentially the core techniques of theoretical condensed matter physics. One needs to calculate the available quantum states using rather sophisticated techniques. Whether the material will become ferromagnetic at low temperature depends on whether the energy of the ferromagnetic state is lower than that of competing states, e.g. antiferromagnetic, superconducting, etc. Calculating the different energies to sufficient accuracy isn't trivial.
(published on 09/07/2011)