After more than a decade of experimental and theoretical studies of the magnetic structure of the fcc Fe/Cu(001) films, most of the researchers agree that the atomic spins of two surface layers are ferromagnetically ordered whereas the magnetic structure of further layers is not ferromagnetic. The conclusions on the magnetic configuration of the nonferromagnetic part vary broadly from a simple collinear layer-by-layer antiferromagnet to complex incommensurate spin-density wave. We report detailed theoretical study of the fcc Fe/Cu(001) films with 6, 7, and 8 ML coverages. The study is based on the density-functional theory calculations with the code allowing for the consideration of arbitrary noncollinear magnetic configurations. We suggest a view on the magnetism of the Fe/Cu(001) films that includes the grouping of the layers into blocks with robust collinear magnetic structure whereas the variation in the relative directions of the moments of different blocks is energetically relatively inexpensive. The robust magnetic structure of the three surface layers was found to be ⇓ ⇑ ⇑. Further layers from the pairs with robust antiferromagnetic structure. We demonstrate that the formation of the robust magnetic blocks corresponds to a certain hierarchy of the interlayer exchange interactions. We suggest that the individual defect pattern of each fabricated film influences before all the relative orientation of the block magnetizations keeping the intrablock structure intact. The calculated inter-atomic exchange parameters are used to study magnetic thermodynamics of the films within mean-field approximation. In particular we focus on possible physical reasons of the appearance of the steplike features in the temperature dependence of the magnetization detected experimentally. We emphasize the important consequences of the inequivalence of the Fe atoms belonging to different layers of the film. This inequivalence makes the properties of the films to be essentially different fromt he properties of the corresponding bulk system. Using a rigid-band model we investigate the dependence of the exchange parameters on the electron number. We critically discuss the possibility of the formation of the incommensurate SDW in the thin Fe/Cu(001) films. This brief formulation of the idea of robust blocking and its experimental confirmation was recently published as a letter [H. L. Meyerheim, J.-M. Tonnerre, L. M. Sandratskii et al., Phys. Rev. Lett. 103, 267202 (2009)].