We present high angular resolution (similar to 2 mas) radio continuum observations of five Seyfert galaxies with flat-spectrum radio nuclei, using the VLBA at 8.4 GHz. The goal of the project is to test whether these flat-spectrum cores represent thermal emission from the accretion disk, as inferred previously by Gallimore et al, for NGC 1068, or nonthermal, synchrotron self-absorbed emission, which is believed to be responsible for more powerful, flat-spectrum nuclear sources in radio galaxies and quasars. In four sources (T0109-383, NGC 2110, NGC 5252, and Mrk 926), the nuclear source is detected but unresolved by the VLBA, indicating brightness temperatures in excess of 10(8) K and sizes, on average, less than 1 pc. We argue that the radio emission is nonthermal and synchrotron self-absorbed in these galaxies, but Doppler boosting by relativistic outflows is not required. Synchrotron self-absorption brightness temperatures suggest intrinsic source sizes smaller than similar to 0.05-0.2 pc, for these four galaxies, the smallest of which corresponds to a light-crossing time of similar to 60 light days or 10(4) gravitational radii for a 108 hi, black hole. In one of these galaxies (NGC 2110), there is also extended (similar to 0.2 pc) radio emission along the same direction as the 400 pc scale jet seen with the VLA, suggesting that the extended emission tomes from the base of the jet. In another galaxy (NGC 4388), the flat-spectrum nuclear source is undetected by the VLBA. We also present MERLIN and VLA observations of this galaxy and argue that the observed, flat-spectrum, nuclear radio emission represents optically thin, free-free radiation from dense thermal gas on scales similar or equal to 0.4 to a few pc. It is notable that the two Seyfert galaxies with detected thermal nuclear radio emission (NGC 1068 and NGC 4388) both have large X-ray absorbing columns, suggesting that columns in excess of similar or equal to 10(24) cm(-2) are needed for such disks to be detectable.