Litotes is when an affirmative is expressed by the negative of the contrary (OED). The OP litotes - not without pride - suggests that the speaker takes pride in the fact that he was not employed as a psychiatrist in camp, or even as a doctor, except for the last few weeks.
The passage also illustrates the problem of saying what you mean by expressing it as a double negative: The meaning is, usually, less clear.
Litotes often implies a subtext which requires context to be understood. Used well, it can be used effectively for a variety of purposes. For example:
"...you didn't look completely out of place"
"...the fact you were drunk didn't make you useless"
to minimise impact;
"...don't worry, it wasn't the worst thing in the world"
to sound modest;
"...I'm not unfamiliar with Beethoven's music"
to highlight how preconceived notions may, in fact, be wrong;
"...this isn't a bad road"
But it is easy to use litotes clumsily and the result can make a speaker sound pompous, ridiculous and falsely modest.