Radical formation through homolytic X-H bond cleavage in LiH, BH, CH4, NH3, H2O, and HF is investigated using natural orbital functional theory in its recent PNOF6 implementation, which includes interelectron-pair correlation, and the results are compared to those of the PNOF5 level of theory, CASSCF wave function methods, and experimental data. It is observed that PNOF6 is able to improve the estimation of the corresponding dissociation energies (De) with respect to PNOF5. When PNOF6 is combined with a better description of the electron pair, through the use of an extended number of coupled orbitals, we obtain further improvements of these quantities. The convergence of the corresponding De values with the number of coupled orbitals is also discussed, finding that a proper convergence of the results is attained with three orbitals. Next, we apply PNOF6 and its improved version PNOF6(3) to describe the thermodynamics of C-H homolytic bond cleavage for a data set of 20 organic molecules in which the C-H bond is broken in the context of different chemical environments. Finally, the radical stabilization energies obtained for such a general data set are compared with the experimental data, demonstrating that the inclusion of interelectron-pair correlation in natural orbital functional theory as in PNOF6 gives a resonable description of radical stability, especially as electron pair description is improved.