Nunca ouvi um falante nativo falar assim. Geralmente, traduções ao pé da letra produzem frases meio estranhas. Também, a frase do DL não representa inglês natural.
I'm a native speaker myself. I would say the "no ... at all" construction is extremely common.
You would actually say "I have no illness..." rather than "I'm not ill/sick"?
Interesting that your speech parallels Portuguese syntax.
Ah, I see what you mean. Yes you're right I would say "I am not sick" or similar to indicate my general state of wellbeing. "I have no illness" would indicate something slightly different - that I am in absolutely A1 condition without even something trivial (I could have a sore toe but still not consider myself sick). That's why i went with the "at all" as the emphasis that nenhum appears often to impart.
Is "I do not have any diseases" correct?
I gave the answer "I do not have any disease" (non native English speaker) and it was rated wrong; only for the missing plural?