Adiabatic electron affinities (AEAs) for the DNA and RNA bases are predicted by using a range of density functionals with a double-zeta plus polarization plus diffuse (DZP++) basis set in an effort to bracket the true EAs. Although the AEAs exhibit moderate fluctuations with respect to the choice of functional, systematic trends show that the covalent uracil (U) and thymine (T) anions are bound by 0.05-0.25 eV while the adenine (A) anion is clearly unbound. The computed AEAs for cytosine (C) and guanine (G) oscillate between small positive and negative values for the three most reliable functional combinations (BP86, B3LYP, and BLYP), and it remains unclear if either covalent anion is bound. AEAs with B3LYP/TZ2P++ single points are 0.19 (U), 0.16 (T), 0.07 (G), -0.02 (C), and -0.17 eV (A). Favorable comparisons are made to experimental estimates extrapolated from photoelectron spectra data for the complexes of the nucleobases with water. However, experimental values scaled from liquid-phase reduction potentials are shown to overestimate the AEAs by as much as 1.5 eV. Because the uracil and thymine covalent EAs are in energy ranges near those of their dipole-bound counterparts, preparation and precise experimental measurement of the thermodynamically stable covalent anions may prove challenging.