In Shakespeare's time, double and even triple negatives could be used to strengthen a negative (see e.g. Is Shakespeare's Double Negative Grammatically Wrong? on English SE). In present-day Standard English, you should not use double negatives unless you want express a positive statement, or intensify the opposite of the negative statement. (An example of the latter would be 'You canNOT NOT go to your father's funeral', as mentioned in a comment by Sibutlasi.)
The disappearance of negative concord in Standard English is sometimes claimed to be a result of the work of prescriptive grammarians in the 18th century, for example Robert Lowth's book A Short Introduction to English Grammar (1762). Negative concord has survived in dialects, see e.g. Negative Concord on the website of the Yale Grammatical Diversity Project.
I wonder if any language with a written record ever went into the opposite direction. Is there any language whose written records show that it acquired negative concord? (There may be languages that acquired it before they started being written down, but in that case, we would need to rely on descriptions of the oral language by users of other languages.)