Okay--first, I agree with milum despite being offended by the remark about ladies that the writer probably meant what he wrote. I found the article, so we can have some context--not that it helps.
http://www.nybooks.com/articles/11940

I have tentatively concluded that somebody somewhere coined this as a word for a kind of "syndrome", as in "there's the equivalent of an a-- in it" syndrome. Possibly a synonym for stupidity.

I just put assinity into the address bar in my internet window, and got "about 19" hits. Here are a couple that led me to this conclusion.
Another example of Tippen's assinity came when... , and some paragraphs down there is Unfortunately, our legal system does not allow a sentence for attorneys for acting like buttheads in court.
http://hometown.aol.com/okamwoc/santee2.htm And,

...commentary on stupidity wherever we find it, from the perspective of a disenfranchised academic insider, viz., me. You'll read my take on current events and academic assinity.
http://www.pitt.edu/~rice/

Thanks, tsuwm; I really enjoyed this "research"!