||A systematic analysis of low-temperature magnetic phase diagrams of Ce compounds is performed in order to recognize the thermodynamic conditions to be fulfilled by those systems to reach a quantum critical regime or, alternatively, to identify other kinds of low-temperature behavior. Based on specific heat (C m ) and entropy results, three different types of phase diagrams are recognized: (i) with the entropy involved in the ordered phase (S MO) decreasing proportionally to the ordering temperature (T MO); (ii) those showing a transference of degrees of freedom from the ordered phase to a non-magnetic component, with their C m (T MO) jumps (Î”C m ) vanishing at finite temperature; and (iii) those ending at a critical point at finite temperature because their Î”C m do not decrease sufficiently with T MO, producing an entropy accumulation at low temperature. Only those systems belonging to the first case, i.e. with S MOâ€‰â†’â€‰0 as T MOâ€‰â†’â€‰0, can be regarded as candidates for quantum critical behavior. Their magnetic phase boundaries deviate from the classical negative curvature below Tâ€‰â‰ˆâ€‰2.5â€‰K, denouncing monotonic misleading extrapolations down to Tâ€‰=â€‰0. Different characteristic concentrations are recognized and analyzed for Ce-ligand alloyed systems. In particular, a pre-critical region is identified where the nature of the magnetic transition undergoes significant modifications, with its âˆ‚C m /âˆ‚T discontinuity strongly affected by the magnetic field and showing an increasing remnant entropy at Tâ€‰â†’â€‰0. Physical constraints arising from the third law at Tâ€‰â†’â€‰0 are discussed and recognized from experimental results.