||A number of specific heat Cm anomalies are reported in Ce- and Yb-lattice compounds around 1 K which cannot be associated to usual phase transitions despite of their robust magnetic moments. Instead of a Cm(T) jump, these anomalies show coincident morphology: (i) a significant tail in Cm/T , with similar power law decay above their maxima ( T>Tm ), (ii) whereas a Cm(T2) dependence is observed below Tm . (iii) The comparison of their respective entropy gain Sm(T) indicates that â‰ˆ0.7R ln2 is condensed within the T>Tm tail, in coincidence with an exemplary spin-ice compound. Such amount of entropy arises from a significant increase of the density of low energy excitations, reflected in a divergent Cm(T>Tm)/T dependence. (iv) Many of their lattice structures present conditions for magnetic frustration. The origin of these anomalies can be attributed to an interplay between the divergent density of magnetic excitations at Tâ†’0 and the limited amount of degrees of freedom: Sm = R ln2 for a doublet-ground state. Due to this â€œentropy bottleneck,â€ the paramagnetic minimum of energy blurs out and the system slides into an alternative minimum through a continuous transition. A relevant observation in these very heavy fermion systems is the possible existence of an upper limit for Cm/TLimTâ†’0 â‰ˆ7 J/mol K 2 observed in four Yb- and Pr-based compounds.