Being eusocial....

Based on today's discussion, it was clear to me that I needed to know more about i) sexual determination in the naked mole rat; ii) something about the organism's genome (link here), and iii) the (perhaps weird and perhaps not) fact that mole rats are extremely resistant to cancer (link here).   So here it is.

Turns out naked mole rats, both males and females are diploid.  In a colony there is one female breeder, and their presence appears to inhibit breeding by other females.
  "Female nonbreeders have small uteri as well as small ovaries without corpora lutea, indicative of complete failure to ovulate. Thus, despite achieving adult body size, the non-reproductive females remain in a pre-pubertal state throughout life." from Holmes et al, 2009.  

If, however, such a non-breeding female is removed from the company of the breeding female, and placed with a male, it can become fertile and breed successfully.  This is how laboratory colonies are started (apparently).

Also, please note that I made a mistake about sex determination in bees. All females are diploid, males are haploid.  Whether a female becomes a female worker or a fertile queen is determined by what it is fed during development.  Female workers can lay eggs, but these are unfertilized and develop into males (drones).  The Wikipedia article on this is good.