Perhaps your sample size is very large or the third level has very small variances. Also, use Tech8 to study how the PSR values progress, seeing how long each iteration takes, and seeing if a particular parameter is often the one with the worst PSR (use Tech1 as well for this purpose).
I have 7186 Individuals in 5186 Households which are nested in 346 villages. On an average I have 1.385 observation from each household and observation from household ranges from 1 to 6.
26% households have one observation, 53% have two and 16% have three (which is cumulatively 95% of the sample). I have calculated the household level ICC which are quite high across the 12 items. But my dilemma is there are mostly 1-3 observation from each household.
What will be the implication of considering the two level model with individual and the village and not accounting for the household?