We present orbital solutions for three low-amplitude spectroscopic binaries discovered in a sample of 20 solar-type IAU radial velocity standard stars observed with the Digital Speedometers at the Harvard- Smithsonian Center for Astrophysics. We update the orbital solutions for HD 114762 and HD 140913, and present a preliminary new solution for HD 29587. For all three orbits, the minimum mass for the secondary is less than 0.08M_{sun}_, the borderline between stellar and substellar masses. We consider the probability that all three binaries have small enough inclination angles so that their companions are above the substellar limit. To do so, we treat the 20 IAU standards as a sample drawn from a population of binaries with a mass-ratio distribution that does not allow any substellar companions. We calculate the probability that such a sample could still have three binaries, with the low-amplitude orbits actually found within the IAU sample. We show that this probability is small, depending on the specific mass-ratio distribution. For example, a flat mass-ratio distribution that assumes there are no substellar companions can be excluded at a high confidence level, 99.7%. We further show that our three detections may imply that the secondary-mass distribution rises near the substellar limit. However, the observations do not yet allow us to distinguish whether the unseen companions of HD 114762, HD 140913, and HD 29587 have stellar or substellar masses. In particular, recent attempts to estimate the mass of the companion of HD 114762 based on assumptions about the intrinsic rotation of the primary are inconclusive, and the companion could easily have a mass as low as 0.02M_{sun}_. We compare our three detections with the null results of four very precise radial velocity searches for substellar companions. The difference is indeed puzzling but can be accounted for if just a small fraction of the solar-type stars, of the order of a few percent, have companions with masses near the substellar limit.