To investigate the relative importance of various small sources of error in theoretical predictions of molecular properties, we report spectroscopic constants for the ground electronic states of BH, CH⁺, and NH which are nearly converged to the adiabatic ab initio limit. Computations are performed using full configuration interaction (FCI) and coupled-cluster singles, doubles and perturbative triples [CCSD(T)] methods with correlation consistent basis sets of double to sextuple-zeta quality. The equilibrium bond lengths, r_e, harmonic vibrational frequencies, omega_e, anharmonicity constants, omega_ex_e, centrifugal distortion constants, D_e, and other quantities are compared with experiment for each species. The systematic dependence of spectroscopic constants on the one-particle basis is used to estimate the complete basis set limit (CBS) values by using a two-point linear extrapolation scheme. The importance of core correlation, scalar relativistic corrections, higher-order electron correlation, and basis set completeness are carefully investigated. Moreover, deviations from the Born-Oppenheimer approximation are studied by computing the diagonal Born-Oppenheimer correction (DBOC). The remaining error is attributed primarily to nonadiabatic effects. Our ab initio limit, adiabatic results for r_e are within 0.0007 angstrom of experiment when nonadiabatic effects are insignificant or have been removed. Adiabatic predictions of omega_e are within 0.5 cm^-1 of experiment.