I'd add that it's important to be consistent, if you're aiming for historical accuracy. That is, you can't have one character using "the N word" while another is referring to African Americans, since the latter designation came much later than your setting. If you trip up and break the illusion of a 1960's setting, you taint the realism that you intend to convey by using the epithet.
Also, if you pepper your text with the word, you risk an air of sensationalism. Additionally, a word in print can have a bigger impact than the same word spoken; if a character repeats a word or phrase or obscenity too often in dialogue, then it will become conspicuous and unnatural, even if a real person might actually use that word/phrase/obscenity just as often.
Consider this less inflammatory example:
George: I was, y'know, processing these TPS Reports and, y'know, I didn't have the cover sheet. So I said to Bill, y'know, that I didn't have them, y'know, and he told me that I better get them or, y'know, I can't turn in my reports at all, y'know? I figured, if that's how it is, y'know, then I'm not going to, y'know, bother with them, especially since they're going to, y'know, downsize me anyway. Y'know?
George might actually speak like that (and, in fact, he
does; believe me). But to see it transcribed that way, the phrase "y'know" becomes obtrusive and cumbersome, pointlessly weighing down the text. Once or twice is all you need to see that George says "y'know" a lot. The same would go for "the N-Word," especially given its inherent cultural/emotional impact.
I had an fiction professor who gave us this excellent piece of advice:
The truth is no excuse for bad fiction. Even if a passage is an accurate portrayal, if it doesn't make good fiction, then it doesn't make good fiction.