I love the way that you worded this question. If I say "no," I'm done. But if I say "yes," I have to explain. I really really want to say no.
But I do believe that Nick Bottom is a fool. And unlike other Shakespeare plays where the foolish character often sees through to the heart of the issue, Nick Bottom is just a clueless fool throughout. He's a terrible actor, but doesn't realize it. In fact, he thinks he awesome. He's not particularly good looking, but assumes that a beauty like Titiana must really be in love with him. He's overbearing among his friends. He doesn't know when to be quiet, and when he does talk, he often says things that just don't make any sense at all. Using act 5 as an example, at one point Bottom says the following:
I see a voice: now will I to the chink,
To spy an I can hear my Thisby's face. Thisby!
Seriously? See a voice and hear a face? Bottom's ridiculousness is all topped off by the fact that he is completely clueless to all of it. It would be one thing if every once in a while a comment of his cut to the heart of an issue, but it just never does. He acts foolish throughout the play, and is therefore a fool.
Bottom is definitely a Fool, and I spell that with a capital 'F' because he is an example of a classical archetype in an ancient, theatrical tradition. Shakespeare wrote different kinds of Fools, some very, very wise, and some apparently "clueless." Bottom is not wise, but he has one of the most endearing of all qualities....Innocence. He is annoying and obnoxious, but everything he says and does comes from enthusiasm and an eagerness to 'play'. He may be foolishly credulous, but he is never cruel or hurtful, and he is kind to those whom he encounters. The most important aspect of Bottom is that he, and he alone, of all the humans in the play, is able to see and consciously interact with the Fairy world. In this way he and his ignorance and his genuine, eager nature are sublimely blessed. He is a Holy Innocent. He is Shakespeare expressing his clarity and deep compassion for the fools in all of us.