Using neutron powder diffraction and Monte Carlo simulations we show that a spin-liquid regime emerges at all compositions in the diamond-lattice antiferromagnets Co (Al1-x Cox) 2 O4. This spin-liquid regime induced by frustration due to the second-neighbor exchange coupling J2 is gradually superseded by antiferromagnetic collinear long-range order (k=0) at low temperatures. Upon substitution of Al3⁺ by Co3⁺ in the octahedral B site the temperature range occupied by the spin-liquid regime narrows and TN increases. To explain the experimental observations we considered magnetic anisotropy D or third-neighbor exchange coupling J3 as degeneracy-breaking perturbations. We conclude that Co (Al1-x Cox) 2 O4 is below the theoretical critical point J2 / J1 =1/8, and that magnetic anisotropy assists in selecting a collinear long-range ordered ground state, which becomes more stable with increasing x due to a higher efficiency of O-Co3⁺ -O as an interaction path compared to O-Al3⁺ -O.