The occurrence of three consecutive positive Indian Ocean Dipole (pIOD) events through 2006-2008, along with the unusual pIOD-La Niña combination in 2007, calls for a statistical assessment into the rarity of such events. To this end, we take 50 years from 19 climate models submitted for the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment (AR4) to provide a 950-year realization. Only six models produce a total of nine pIOD-La Niña pairs, with one occurrence of three consecutive pIOD events. The rarity is not due to an overly strong model El Niño-pIOD coherence. Two models produce the 2006-2007 sequence of one pIOD with an El Niño followed by a pIOD in a La Niña year, however, no models simulate the observed mechanism. Although the triggering processes vary between models, a commonality is that the trigger resides in the ocean, highlighting the importance of subsurface ocean observations in predicting pIODs.