The bond dissociation energies (0,) of C>H>, &Ho, CzHa, Nz, N2H2, and NzH4 are studied at various levels of correlation treatment. The convergence of 0. with respect to the one-particle basis is studied at the single-reference modified coupled-pairfunctional (MCPF) level. At all levels of correlation treatment, the errors in the bond dissociation energies increase with the degree of multiple bond character. The multireference configuration-interaction (MRCI) D, values, corrected for an estimate of higher excitations, are in excellent agreement with those determined using the size-extensive averaged-coupled-pair-functional (ACPF) method. We find that the full-valence complete-active-space self-consistent-field (CASSCF) /MRCI calculations are reproduced very well by MRCI calculations based on a CASSCF calculation that includes in the active space only those electrons involved in the C-C or N-N bonds. To achieve chemical accuracy ( 1 kcal/mol) for the D, values of the doubly bonded species C,H, and N,H, requires one-particle basis sets including up through h angular momentum functions (I= 5) and a multireference treatment of electron correlation: still higher levels of calculation are required to achieve chemical accuracy for the triply bonded species CzH2 and Nz,