There is clear observational evidence that some narrow-line (type 2) active galactic nuclei (AGNs) have a hidden broad-line region (BLR) and are thus intrinsically broad-line (type 1) AGNs. Does this AGN unification apply for all type 2 AGNs? Indirect arguments suggest that some "true" type 2 AGNs, i.e., AGNs having no obscured BLR, do exist, but it is not clear why the BLR is missing in these AGNs. Here we point out a possible natural explanation. The observed radius-luminosity relation for the BLR implies an increasing line width with decreasing luminosity for a given black hole mass (MBH). In addition, there appears to be an upper limit to the observed width of broad emission lines in AGNs of Δvmax ~ 25,000 km s-1, which may reflect a physical limit above which the BLR may not be able to survive. Thus, at a low enough luminosity the BLR radius shrinks below the Δvmax radius, leaving no region where the BLR can exist, although the AGN may remain otherwise "normal." The implied minimum bolometric luminosity required to sustain a BLR with Δv < 25,000 km s-1 is Lmin ~ 1041.8(MBH/108 M☉)2. All AGNs with L < Lmin are expected to be "true" type 2 AGNs, i.e., narrow-line AGNs without a hidden BLR. Predictions for the true nature of low-luminosity AGNs in two samples of nearby galaxies are provided. These can be used to test the above Lmin conjecture and the predictions of other models for the size and origin of the BLR.