James was the son of John Oliver and Zerilda Dunn.
"United States Census, 1870," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MXW2-RL1 : accessed 3 September 2015), J T Oliver in household of J P Oliver, Kentucky, United States; citing p. 45, family 307, NARA microfilm publication M593 (Washington D.C.: National Archives and Records Administration, n.d.); FHL microfilm 545,983.
"United States Census, 1880," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MCZ8-79D : accessed 3 September 2015), James L Oliver in household of John P Oliver, Parkersville, Lyon, Kentucky, United States; citing enumeration district 144, sheet 67B, NARA microfilm publication T9 (Washington D.C.: National Archives and Records Administration, n.d.), roll 0430; FHL microfilm 1,254,430.
https://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=160913424
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To confuse the two would be very careless.