A number of specific heat C m 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 C m (T ) jump, these anomalies show coincident morphology: (i) a significant tail in C m /T , with similar power law decay above their maxima (T > T m ), (ii) whereas a C m (T 2 ) dependence is observed below T m . (iii) The comparison of their respective entropy gain S m (T ) indicates that ≈ 0.7Rln2 is condensed within the T > T m 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 C m (T > T m )/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 T → 0 and the limited amount of degrees of freedom: S m = Rln2 for a doublet-ground state. Due to this "entropy bottleneck," 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 C m /T LimT →0 ≈ 7 J/mol K 2 observed in four Yb-and Pr-based compounds.