On Thursday the 58-year-old accountant was granted permission to seek a judicial review of the decision to refuse him access under the Freedom of Information Act 2000.
Robert was raised in Kenya as the son of Cynthia and Douglas Brown. Records show that he was born on January 5, 1955, in the Kenyan capital Nairobi. His birth was not registered until February 2.
Later that year the Princess called off her wedding to Peter Townsend.
Robert thinks his father may be Scottish aristocrat Robin Douglas-Home. His theory is that Margaret's pregnancy was covered up using body doubles. His mother Cynthia worked as a model for Hardy Amies, a favourite designer with both the Queen and her sister.
He argued that Buckingham Palace working with the Attorney General arranged to keep the contents of Margaret's will secret because it carried details of his birth. She died in 2002.
Lawyers acting for the royal family have previously dismissed his case as that of "a fantasist seeking to feed his private obsession".
He told The Guardian that his parents were distant towards him in comparison to the way they treated his siblings. Sometimes his birthday was forgotten and they never talked about his birth.
This week he said: "I am delighted that the points I have raised have been recognised as arguable."
Of his claim to royal parentage, he added: "Hopefully I am not a nutcase. I am either right or I am wrong."