The bond dissociation energies (D(e)) of C2H2, C2H4, C2H6, N2, N2H2, and N2H4 are studied at various levels of correlation treatment. The convergence of D(e) with respect to the one-particle basis is studied at the single-reference modified coupled-pair-functional (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(e) 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(e) values of the doubly bonded species C2H4 and N2H2 requires one-particle basis sets including up through h angular momentum functions (l = 5) and a multireference treatment of electron correlation: still higher levels of calculation are required to achieve chemical accuracy for the triply bonded species C2H2 and N2.