We have studied the magnetic phase diagram for the dilute alloys (Dy_xY_1-x)Al₂ in the concentration range 0.10<or=x<or=0.50. We find the systems to be disordered magnets with random magnetic anisotropy. For x<or=0.30, the system undergoes a single transition from paramagnetic (PM) to spin-glass (SG) state. For 0.35<or=x<or=0.50, a re-entrant coherent spin-glass (CSG) phase appears between the PM and SG states, formed by intrinsic nearly ferromagnetic domains. The triple point is located at x approximately=0.35. However, no evidence of any long-range magnetic order was found in the concentration range studied. We have also determined the temperature variation of the spontaneous Edwards-Anderson order parameter for x<or=0.30. The scaling analyses performed for the non-linear susceptibility indicate true phase transitions for the PM-SG line for x<0.35, and yield values for the critical exponents beta , gamma and delta .