I don't know that there is a connection but the names certainly are simular.
I have a John Mitchell Bright, b. in Ms who married Sarah Wall. I have only 2 children for them, John Mitchell & Ann Bright. This John Mitchell Bright was the son of George Henry Bright, b. March 08, 1824 in Walton Co, Fla., d. between 1894 - 1900 in Miss and Julia A. Mitchell, b. ca. 1826 in Washington Co, Fla, m. May 18, 1859 in Mobile, Al.
If you think there might be a match, let me know, I have more. Most of mine is later generations, not currant ones.