The relationship between magnetism and morphology of pseudomorphic ultrathin films of fcc Fe on Cu(001) is reexamined in view of morphological data obtained with scanning tunneling microscopy. Two types of films showing different magnetic properties were investigated. Films prepared at 300 K are found to grow in a good layer-by-layer mode. Their magnetism appears to be influenced by structural instabilities connected with a martensitic fcc to bcc transformation. Films deposited at 130 K and later annealed at 300 K grow in a Stranski-Krastanov-like mode, resulting in a drastically different surface morphology. They are found to be structurally stable in the region where the spin reorientation takes place. An attempt is made to correlate the enhanced roughness of these films with their magnetic properties. Finally, the connection between the morphological percolation and the onset of long-range ferromagnetic ordering is discussed. The percolation of the first layer is found to be insufficient to establish long-range ferromagnetic ordering in these films.