Well I do it, but then I spend a lot of time as a mathematician.

If it gets confusing I think using alternative bracket glyphs assists ([{<« »>}]).

[Though using the guillemets (« ») as brackets can get you in to trouble, as a lot of languages use them as speech marks.^(Wikipedia)]

I believe it's acceptable, but vaguely considered poor form, and I tend to avoid it (often by restructuring a sentence and busting out some emdashes) unless I'm intentionally using it to be cute.

My English Composition professor told us that if you ever feel you need to use nested parentheses, that is one sure sign you need to rewrite the sentence instead.

Here's what one technical editor has to say on the topic:

Nesting parentheses should be done with square brackets "([ ])". But since square brackets are usually used for citations, this is confusing. The answer is to eliminate all nested parentheses by appropriate rewording. (Dashes, otherwise not recommended, can have a use here.)

I have found that people who appreciate stressing thoroughly the logical construction of their sentences to make them totally unambiguous tend to use parentheses a lot, and to nest them, even though (see other answers) it's considered poor form. In particular, scientists (especially mathematicians and logicians) seem to do that more commonly than other people.

In looking at this question, I was immediately reminded of the work of William Faulkner, an undoubtedly well-known author in the United States. He is notorious for his complex sentences that can go on for pages. In some of them, he unapologetically uses nested parentheses.

While Faulkner sets a precedent doing this, it is not at all proof of the 'correct' usage. His motivation is quite different. I like the way Louis Rubin puts it in his essay The Dixie Special: William Faulkner and the Southern Literary Renascence:

His very style itself, with the long sentences, the liberal deployment of adjectives, the parentheses, and the parentheses within parentheses, proceeds from the conviction that reality can be represented only when presented in its full complexity, leaving out nothing that is important. His admiration for Thomas Wolfe, he said on several occasions, was for that novelist's attempt, however impossible of fulfillment, to put all experience on the head of a pin.

I bring Faulkner's example up for the sake of introducing a different angle on the OP's question. It may not be the best choice stylistically to nest parentheses, but there have been literary precedents that show the practice. The example of Faulkner's usage of the nested parentheses shows that there is a purpose in his writing to convey a sense of complexity and to leave nothing out.

Parentheses are a way to stuff more ideas into a sentence than it could otherwise bear. They make life easier for the writer who is trying to capture all his ideas as they bubble up, but harder for the reader trying to make sense of it all. If you nest your parentheses, you risk losing your reader entirely.

So I'd say, in your first draft, go ahead and use as many parentheses as you want, but in later drafts you should try to eliminate them — especially the nested ones.