Type 2 AGNs with intrinsically weak broad emission lines (BELs) would be exceptions to the unified model. After examining a number of proposed candidates critically, we find that the sample is contaminated significantly by objects with BELs of strengths indicating that they actually contain intermediate-type AGNs, plus a few Compton-thick sources as revealed by extremely low ratios of X-ray to nuclear IR luminosities. We develop quantitative metrics that show two (NGC 3147 and NGC 4594) of the remaining candidates to have BELs 2-3 orders of magnitude weaker than those of typical type-1 AGNs. Several more galaxies remain as candidates to have anomalously weak BELs, but this status cannot be confirmed with the existing information. Although the parent sample is poorly defined, the two confirmed objects are well under 1% of its total number of members, showing that the absence of a BEL is possible, but very uncommon in AGN. We evaluate these two objects in detail using multi-wavelength measurements. They have little X-ray extinction with N_H < 10^21 cm^{-2}. Their IR spectra show strong silicate emission (NGC 4594) or weak aromatic features on a generally power law continuum with a suggestion of silicates in emission (NGC 3147). No polarized BEL is detected in NGC 3147. These results indicate that the two unobscured type-2 objects have circumnuclear tori that are approximately face-on. Combined with their X-ray and optical/UV properties, this behavior implies that we have an unobscured view of the nuclei and thus that they have intrinsically weak BELs. We compare their properties with those of the other less-extreme candidates. We then compare the distributions of bolometric luminosities and accretion rates of these objects with theoretical models that predict weak BELs.
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