We present photometry and spectroscopy of the Type IIn supernova (SN) 2011ht, identified previously as a possible SN impostor. The light curve exhibits an abrupt transition from a well-defined ̃120 d plateau to a steep bolometric decline, plummeting 4-5 mag in the optical and 2-3 mag in the infrared in only ̃10 d. Leading up to peak brightness (M$_V$ = -17.4 mag), a hot emission-line spectrum exhibits strong signs of interaction with circumstellar material (CSM), in the form of relatively narrow P-Cygni features of H I and He I superimposed on broad Lorentzian wings. For the latter half of the plateau phase, the spectrum exhibits strengthening P-Cygni profiles of Fe II, Ca II and Hα. By day 147, after the plateau has ended, the SN entered the nebular phase, heralded by the appearance of forbidden transitions of [O I], [O II] and [Ca II] over a weak continuum. At this stage, the light curve exhibits a low optical luminosity that is comparable to that of the most subluminous Type II-P supernovae, and a relatively fast visual wavelength decline that appeared to be significantly steeper than the $^56$Co decay rate. However, the total pseudo- bolometric decline, including the infrared luminosity, is consistent with $^56$Co decay, and implies a low $^56$Ni mass in the range 0.006-0.01 M$_☉$, near the lower end of the range exhibited by SNe II-P. We therefore characterize SN 2011ht as a core-collapse SN very similar to the peculiar SNe IIn 1994W and 2009kn. These three SNe appear to define a subclass, which are Type IIn based on their spectrum, but that also exhibit well-defined plateaus and produce low $^56$Ni yields. We therefore suggest Type IIn-P as a name for this subclass. The absence of observational signatures of high-velocity material from SNe IIn-P could be the result of an opaque shell at the shocked SN-CSM interface, which remains optically thick longer than the time-scale for the inner ejecta to cool and become transparent. Possible progenitors of SNe IIn-P, consistent with the available data, include 8-10 M$_☉$ stars, which undergo core collapse as a result of electron capture after a brief phase of enhanced mass loss, or more massive (M gtrsim 25 M$_☉$) progenitors, which experience substantial fallback of the metal-rich radioactive ejecta. In either case, the energy radiated by these three SNe during their plateau (2-3 × 10$^49$ erg for SN 2011ht) must be dominated by CSM interaction, and the subluminous tail is the result of low $^56$Ni yield.