Double negative: Nobody did not speak

Can one say:

[color=red]1-Nobody did not speak.
instead of:
[color=indigo]Everybody spoke.
Nobody managed not to speak.

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No; that’s why we eschew double negatives.
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“Nobody did not speak,” is an awkward but possible way of saying that everyone spoke, but you’ve have to stress the word “not”. It would be expressed this way only in rare situations.

“Didn’t nobody speak,” is colloquial dialect and means that nobody spoke.

Double negatives are tricky, because it’s hard to tell if someone is using them with standard English grammar (and correct stress) to indicate a positive, or if the person is speaking low colloquial English and using them to indicate a negative.