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, re, harmonic vibrational frequencies, omegae, anharmonicity constants, omegaexe, centrifugal distortion constants, De, 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 re are within 0.0007 angstrom of experiment when nonadiabatic effects are insignificant or have been removed. Adiabatic predictions of omegae are within 0.5 cm-1 of experiment.