Competing Jahn-Teller distortions combined with geometrical frustration give rise to a rich phase diagram as a function of x(Cu) and temperature in the spinel system Ni1-xCuxCr2O4. The Jahn-Teller distortion of the end members acts in opposite ways, with an elongation of the NiO4 tetrahedra resulting in a structural transition at T-S1 = 317K in NiCr2O4, but a flattening in the CuO4 tetrahedra at T-S1 = 846K in CuCr2O4. In both cases the symmetry is lowered from cubic (Fd (3) over barm) to tetragonal (I4(1)/amd) on cooling. In order to follow the influence of Jahn-Teller active Ni2+ and Cu2+ ions on the structural and magnetic properties of chromium spinels, we have investigated a series of samples of Ni1-xCuxCr2O4 by x-ray and neutron powder diffraction. In the critical range 0.10 < x(Cu) < 0.20, strong orthorhombic distortions were observed, where competing Jahn-Teller activities between the Cu2+ and Ni2+ ions result in distortions along both the a and c axes. For Ni0.85Cu0.15Cr2O4, the orthorhombic structure (Fddd) is stabilized up to T-S2 = 368(2) K, close to the first structural phase transition at T-S1 = 374(2) K. A ferrimagnetic spin alignment of the Ni/Cu and chromium atoms sets in at much lower temperature TC = 95K in this compound. The end members NiCr2O4 and CuCr2O4 undergo this ferrimagnetic transition at TC = 74 and 135 K, respectively. These transitions are accompanied by the structural change to the orthorhombic symmetry which relieves the frustration. NiCr2O4 and Ni0.85Cu0.15Cr2O4 undergo a second magnetic transition at TM2 = 24 and 67K due to a superimposed antiferromagnetic ordering of the Cr moments resulting in a noncollinear magnetic structure. In the system Ni1-xCuxCr2O4, the magnetic transitions TC and TM2 merge with increasing copper content up to x(Cu) similar to 0.5. For the Ni-rich chromites, geometrical frustration causes a strong reduction of the chromium moments, where magnetic long-range order coexists with a disordered spin-liquid-like or a reentrant-spin-glass-like state. This paper provides insight into the interplay between the Jahn-Teller effect, geometrical frustration, and long-range magnetic order in these complex systems.