In that story, the percentages are meant to be satirical. Twain's drawing attention to the fact that really, there's no difference between the slave and the enslaver.
First, it's worth noting that the percentage of African ancestry doesn't always represent the phenotype, since genetic expression is a complex thing. There are hundreds of accounts of quadroon slaves (1/4 black) who were reported to be whiter looking than most white people, whiter-looking than their own owners. There are a lot of reports from that era of people complaining - not about slavery, but about white slavery because these mulatto slaves looked too completely white.
As for how low one's black ancestry could go and still be sold as slave, there are plenty of examples of 1/16 and 1/32. Or even lower.
For example, Reverend Calvin Fairbank, a methodist minister and abolitionist from New York wrote about a girl named Eliza who was 1/64 black, and whom he purchased at a slave auction in Kentucky:
I only remember that it was early in May, 1843, that my sympathy and patriotism were roused in behalf of one of the most beautiful and exquisite young girls one could expect to find in freedom or slavery. She was the daughter of her master, whose name I withhold for laudable reasons, and was as free of African blood as Kate McFarland, being only one sixty-fourth African. She was self-educated, and accomplished in literature and social manners, in spite of the institution cursing her race ; and her heartless, jealous mistress had doomed her to be sold on the block, hating her for her beauty and accomplishments.
Eliza had been confined in an upper room of the Lexington jail. She recognized me as I was walking in the jail-yard, and drew my attention by tapping upon the window. I called upon her in her room, learned her situation, and hastened to Cincinnati to Levi Coffin, then with him to Hon. S. P. Chase, Nicholas Longworth, Samuel Lewis and others, returning to Lexington with twenty-two hundred and seventy-five dollars, and a paper authorizing me to draw twenty-five thousand if necessary to save the girl.
He goes on to describe that when she was put on the auction block, he got into a bidding war with a Frenchman who coveted the girl, but he outbid the Frenchman and was able to buy Eliza for $1485. Eliza was then freed and taken to Cincinnati, where she eventually married and lived as a white woman. Only a few people knew she had once been a slave.
He mentions that her master (who was also her father) was glad that she had been freed, and never intended to sell her on the auction block but was forced to do so by his wife. He contributed $100 towards the money that went into buying and freeing her.
I haven't come across any statistics of how many slaves had 1/16 or 1/32 or 1/64 African ancestry, but there were certainly some. It would have been pretty inconsequential just how much African ancestry they had, because legally the one-drop rule or maternal inheritance didn't care about fractions, and appearance-wise it didn't take much to make the person look white. As lots of people have reported, there were many, many quadroons that looked clearly and unambiguously white but were kept and sold as slaves. The "fancy trade", which was especially prominent in New Orleans and Kentucky specialized in white-looking young women who were sold for prostitution in bordellos or as mistresses to the wealthy.
Edit: The extract I quoted above is from the book "Rev. Calvin Fairbank during Slavery Times: How he fought the good fight to prepare the Way" which was published by R.R. McCabe & Co. in Chicago in 1890.
Not uncommon. There is a very striking picture of some emancipated slaves that appeared in Harper's magazine in 1864, with an explanation of who the emancipated slaves were. A couple of them were some kids who are white, but were slaves. The photo and description was designed to bring sympathy to the abolitionist movement by making people feel empathy with the white slaves, even if they couldn't identify with the black slaves.
Pic here: http://www.frantzkebreau.com/uploads/5/4/8/7/5487562/2248303_orig.jpg?248
The description of some of the people in the photos:
To the Editor of Harper's Weekly:
THE group of emancipated slaves whose portraits I send you were brought by Colonel Hanks and Mr. Philip Bacon from New Orleans, where they were set free by General Butler. Mr. Bacon went to New Orleans with our army, and was for eighteen months employed as Assistant-Superintendent of Freedmen, under the care of Colonel Hanks. He established the first school in Louisiana for emancipated slaves, and these children were among his pupils. He will soon return to Louisiana to resume his labor.
REBECCA HUGER is eleven years old, and was a slave in her father's house, the special attendant of a girl a little older than herself. To all appearance she is perfectly white. Her complexion, hair, and features show not the slightest trace of negro blood. In the few months during which she has been at school she has learned to read well, and writes as neatly as most children of her age…
ROSINA DOWNS is not quite seven years old. She is a fair child, with blonde complexion and silky hair. Her father is in the rebel army. She has one sister as white as herself, and three brothers who are darker…
CHARLES TAYLOR is eight years old. His complexion is very fair, his hair light and silky. Three out of five boys in any school in New York are darker than he. Yet this white boy, with his mother, as he declares, has been twice sold as a slave. First by his father and "owner," Alexander Wethers, of Lewis County, Virginia, to a slave-trader named Harrison, who said them to Mr. Thornhill of New Orleans. This man fled at the approach of our army, and his slaves were liberated by General Butler. The boy is decidedly intelligent, and though he has been at school less than a year he reads and writes very well. His mother …
Harper’s Weekly, January 30 1864
