Are double negatives ever appropriate in English?

Solution 1:

Double negatives can be perfectly fine in English.

• If their sum is supposed to be negative, double negatives are very informal or slang in modern English. This usage is frowned upon by many people even if used in speech, unless ironically.

1.) I don't see nothing. (= I don't see anything.)

• If their sum is supposed to be positive, it is generally acceptable in all registers:

2.) I suppose that is not impossible. However, it seems far fetched.

Here the double negative expresses a weak positive, a very common construction.

3.) Not bad, not bad at all! You have just saved her life, young man.

This is a figure of speech called litotes: the double negative (if that's what it is) is used to express a strong positive. Sometimes any double negative with a positive meaning is considered a litotes, including the unremarkable example 2 above. Other people restrict the term to those negations that express a strong positive through an apparently weak positive, in a mildly ironical manner, as in this example (3).

4.) Never a day goes by that I do not miss her. (= I always miss her.)

This is the rhetorical double negative, often considered a form of litotes. It expresses a strong positive, though without irony.

5.) Well, I didn't not enjoy it, but...

Here the word not is used twice, once in contracted form (don't) and once in full, to express a weak positive. Double not is a special case: it is felt to be even more redundant than other double negatives and sounds rather colloquial. The majority will probably use this in speech and informal writing, where it is perfectly acceptable, but not elsewhere.

The boundary between negations and other kinds of words is by no means sharp. The prefixes un- and im-/in- are usually considered negatives, as are hardly and many others; bad is sometimes considered a negative word, sometimes not, etc.

Solution 2:

There are two types of double negative, and I think it is important to recognize the difference between the two. They are:

1. negative concord, which is standard in Spanish, French, and many other languages, where negation amplifies with the addition of more negative words (or they are simply required). When people say "double negatives are not part of Standard English", they are talking about negative concord. (It is used in a number of English dialects, as well as in certain informal registers in most dialects.)
2. two negatives yielding a positive, which is what you appear to be talking about. This exists even in Standard English. Your particular example might be a bit awkward, but that is just because you wrote an awkward sentence, not because double negatives are bad. Other double negations sound perfectly fluid, for example:

It's not that I didn't enjoy it...

or:

I'm not unhappy/dishonest.

or:

When I see someone in trouble, I can't not help.

I think everyone would agree that these are common Standard English constructions.

(However, as I said, even your awkward example is not ungrammatical in Standard English.)

Side note: even negative concord exists in Standard English in certain cases. For example, in response to negative questions:

"You're not coming?" —  "No (I'm not)."

Essentially, when someone asks a negative question, the answer of "no" does not yield the positive, i.e. it does not mean "No, I am coming". Instead, "no" creates negative concord, maintaining the negation of the question.

In other languages where they don't employ this negative concord, like Japanese for example, the way to respond to "you're not coming" would be "yes, I am not coming". Answering "no" would would mean "I am coming".