This paper reports the latest results from a near-infrared search for hidden broad-line regions (BLRs: FWHM >~ 2,000 km/s) in ultraluminous infrared galaxies (ULIGs). The new sample contains thirty-nine ULIGs from the 1-Jy sample selected for their lack of BLRs at optical wavelengths. The results from this new study are combined with those from our previous optical and near-infrared surveys to derive the fraction of all ULIGs with optical or near-infrared signs of genuine AGN activity (either a BLR or [Si VI] emission). Comparisons of the dereddened emission-line luminosities of the optical or obscured BLRs detected in the ULIGs of the 1-Jy sample with those of optical quasars indicate that the obscured AGN/quasar in ULIGs is the main source of energy in at least 15 -- 25% of all ULIGs in the 1-Jy sample. This fraction is 30 -- 50% among ULIGs with L_ir > 10^{12.3} L_sun. These results are compatible with those from recent mid-infrared spectroscopic surveys carried out with ISO. (abridged)

40 pages including 10 figures and 3 tables (Table 3 should be printed in landscape mode)