>govt.

So far in this thread I've only seen mention of:

   - INcompetent and nice
   - competent and NOT nice

... also, with the qualifier that both extremes seem to be minorities "on the team" however defined (Gauss comes to mind here).

But I as well as family and friends have also encountered the other variations, and more often than rarely. Incidentially a close relative has just had this experience in a public sector job (Health Care, Europe):

Person in question is

   - competent and nice (as regards to her field of work and work ethic)

Person got hired to a small unit in which everybody else seemed to be:

   - INcompetent and NOT nice

... except for the manager who was one of the types discussed here:

   - INcompetent and nice

The work environment was fully toxic and not long after enrolling my competent and nice relative was on sick leave as a nervous wreck, and being laid off by the nice manager, because, as he said:

- "I'll have to fire either You or the full rest of the team, but as I need hands I have only one option."

(I have seen flavours of this happening in big private enterprises too, so it's not reserved for public/govt. jobs IMHO)