The reliability of density functional theory and other electronic structure methods is examined for anharmonicities and spectroscopic constants of the ground electronic states of several diatomic molecules. The equilibrium bond length r(e), harmonic vibrational frequency omega (e), vibrational anharmonicity omega (e)x(e), rotational constant B-e, centrifugal distortion constant (D) over bar (e), and vibration-rotation interaction constant alpha (e) have been obtained theoretically for BF, CO, N-2, CH+, and H-2. Predictions using Hartree-Fock, coupled-cluster singles and doubles (CCSD), coupled cluster singles and doubles with perturbative triples [CCSD(T)], and various density functional methods (S-VWN, BLYP, and B3LYP) have been made using the 6-31G(*), aug-cc-pVDZ, and aug-cc-pVTZ basis sets and compared to experimental values. Density functional theory predictions of the spectroscopic constants are reliable (particularly for B3LYP) and often perform as well as the more expensive CCSD and CCSD(T) estimates.