Neutron-diffraction and direct magnetic measurements have been used to determine the magnetic structures of platinum + iron alloys in the neighbourhood of the composition Pt$_3$Fe. It is found that the magnetic structure is sensitively dependent on the precise chemical order, in particular on the number of nearest-neighbour iron atoms. Alloys of the stoichiometric composition, where the chemical order is perfect, have a simple antiferromagnetic structure, but addition of excess iron leads to the coexistence of two different structures, with different Neel temperatures but a single, undistorted, cubic unit cell. An incipient ferromagnetic state which shows a cluster type of behaviour is associated with the presence of excess iron. Increase of iron content beyond 34% leads to the growth of bulk ferromagnetism. Throughout the sequence of structural changes the iron atoms carry a magnetic moment of about 3.3 $\mu_B$ and there appears to be very little alined moment on the platinum atoms.