I am not sure if he is related to anyone in Fancy Hill, but James K. Polk Cockburn (Cogburn) lived in Montgomery County, Arkansas.However, he and his family moved from Arkansas to Texas, as you can see by James' census records:
1850 Census in Gap, Montgomery, Arkansas: "James Polk Cockburn" (in the household of his parents, C. T. and Sarah Cockburn)
1860 Census in Sulphur Spring, Montgomery, Arkansas: "Jane Cockburn" (in the household of his widowed mother, Sarah Cockburn) [Note: The first name is a census taker error. It should be James.]
1880 Census in Precinct 5, Wise, Texas: "James P. Cockburn"
1900 Census in Justice Precinct 2, Hood, Texas: "James N. Cogburn" [Note: My theory is that James K. Polk Cockburn's (Cogburn) full name was James Knox Polk Cockburn after the President; therefore, the census taker took the name Knox to begin with an "N" instead of a "K."]
1920 Census in Justice Precinct 4, Red River, Texas: "J. G. Cogburn" (in the household of his daughter, "Ms. Allie Hastings") [Note: For some reason, James K. Polk Cockburn went by J. G. Cogburn in his later years.In fact, he signed a letter to his son Newt Cogburn this way.]
1930 Census in Precinct 4, Red River, Texas: "James D. Cogburn" [Note: The middle initial is a census taker error.]
James K. Polk Cockburn's father, as shown by the 1850 census, was Clark T. Cockburn, who was born 11 Aug 1812 in Franklin Co., Georgia, and died 26 May 1856 in Montgomery Co., Arkansas.Clark's father was George Cockburn, who was born circa 1746 in England and died before 3 Nov 1834 in Franklin Co., Georgia.