||The evolution of the behavior of the Ce 4f electron with the Cu content in the alloy Ce(Pd1-xCux)(2)Si-2 was investigated by determining its structural, magnetic, transport, and thermal properties. Results obtained on 15 samples with compositions distributed along the whole concentration range allow us to draw a magnetic phase diagram showing three distinct regions. The first region at low Cu concentrations corresponds to the antiferromagnetic-ordered regime. The fast suppression of this magnetic order between x = 0.2 and x = 0.25 suggests the presence of a critical point which separates this region from the second one at larger.x values where no magnetic order is observed. The main characteristic of this second region, which extends up to x = 0.7, is a pronounced increase of the Kondo temperature T-K, from T-K approximate to 15 K at x = 0.25 to T-K approximate to 50 K at x = 0.7. This is in contrast with the fast decrease of T-K in the region with x > 0.7. In this region a small anomaly with a ferromagnetic character is observed in the specific hear arround 4 K. The change in the evolution of T-K with x coincides with a change in the composition dependence of the c lattice parameter of the tetragonal structure, pointing to a structure-related origin of this unusual T-K(x) behavior. This nonmonotonous T-K(x) dependence is reproduced using local-density-approximation band-structure calculations, which show a maximum of the hybridization strength between conduction electrons and Ce 4f stares for intermediate Cu concentrations. Such a T-K(x) dependence which contradicts the monotonous dependence observed in all other Ce-based alloy systems is the hallmark of the Ce(Pd1-xCux)(2)Si-2 alloy, which cannot be described as a Fermi-Liquid system in the nonmagnetic region. [S0163-1829(98)02118-3].