Nissan, Nissim and Natan are 3 different names.
Nissan ניסן is usually someone who was born during that month (or named for someone who was born in Nissan).
Natan נתן - in English transliterated as Nathan - pronounced in Ashkenazic pronunciation as Nossen or Noosen.
Nissim ניסים or נסים which translates as miracles.
There are no rules, but Nissan is more likely to be of Ashkenazic descent, and Nissim - Sephardic.