Converging lower bound to the critical screening parameterD_c associated with the ground state of a two-particle system interacting through a cut-off Coulomb potential is obtained analytically using an improved condition for the absence of bound states. The predicted numerical result for the lower bound is found to be within 10⁻³% of the exact result. On the other hand, a multi-parameter variational approach yields a tight upper bound, within 0.54% of the exact result. It is shown that the critical screening parameter for the exciteds-states can also be determined in an approximate way. We obtainD_c^ms ≈ [0.764435n⁻²+0.617737n⁻³]⁻¹ wheren is the principal quantum number. The predictedD_c for various quantum states (n=1 to 8) are in good agreement with the values obtained numerically by Singh and Varshni.