Learn English – Is “cannot not say” standard English

The reason you were told it’s a mistake is because it is a usage found in dispreferred dialects. Similar to use of ain’t, double negation—also known as negative concord—is quite common in many dialects of English, but it is not part of any formal register of English—that is to say, it is not a feature of the standard English taught in schools. Negative concord is a feature of the standard dialect of many languages, such as Spanish—just not standard English.

“I can’t get no satisfaction” in formal standard English would be “I can’t get any satisfaction” or “I can get no satisfaction”. However, it’s important to note that although formal modern English does not have a negative concord rule, such a rule is not inherently “illogical”, as many commenters who try to explain English’s rule claim.

Here is a post on Language Log about “overnegation”.

That isn't double-negation, that's single negation. Double-negation would be this:

That book does not have no pages.

Sentences like the above are correct in many languages and are used in many non-standard forms of English, but are not correct in standard English.

You may be confused by the word any, which is a negative polarity item. The word any is not itself a negative, but it is required in place of no or some when the main verb is negative.