Models of the present-day intrinsic population of cataclysmic variables predict that 99 per cent of these systems should be of short orbital period (P_orb<~2.5 h). The Galaxy is old enough that ~70 per cent of these stars will have already reached their orbital period minimum (~80 min), and should be evolving back toward longer periods. Mass-transfer rates in these highly evolved binaries are predicted to be <~10^-11 Msolar yr^-1, leading to M_V of ~10 or fainter, and the secondaries would be degenerate, brown dwarf-like stars. Recent observations of a group of low-luminosity dwarf novae (TOADs) provide observational evidence for systems with very low intrinsic M_V and possibly low-mass secondaries. We carry out population synthesis and evolution calculations for a range of assumed ages of the Galaxy in order to study P_orb and _solarM distributions for comparison with the TOAD observations. We speculate that at least some of the TOADs are the predicted very low-luminosity, post-period-minimum cataclysmic variables containing degenerate (brown dwarf-like) secondaries having masses between 0.02 and 0.06 Msolar and radii near 0.1 Rsolar. We show that these low-luminosity systems are additionally interesting in that they can be used to set a lower limit on the age of the Galaxy. The TOAD with the longest orbital period currently known (123 min), corresponds to a Galaxy age of at least 8.6x10^9 yr.