No. This is not some Time-Turner trick. Harry Potter was not his own great-grandfather. That would have been, well, WEIRD. Read J.K. Rowling’s explanation below for more about Harry’s ancestor Henry Potter:
Henry Potter (Harry to his intimates) was a direct descendant of Hardwin and Iolanthe and served on the Wizengamot from 1913 – 1921.
This except was brought to you by the new article from J.K. Rowling about the Potter family. There’s so much more than this story about Henry Potter. This essay tells us how the Potter family name originates with the wizard Linfred of Stinchcombe, whose nickname was “the Potterer” (a name that eventually was transformed to “Potter”), and whose claim to fame was the invention of Skele-gro. Ironically, Harry had to use his ancestor’s potion to regrow his arm bones after Professor Lockhart inadvertently removed them after a Quidditch match during Chamber of Secrets.
This invention is a major reason why the Potter family was so wealthy. That, and Linfred’s eldest son Hardwin married Ignotus Peverell’s eldest granddaughter—Iolanthe—which brought the Invisibility Cloak into the Potter family. “From this time on,” JKR’s article states, “the cloak was handed down to the eldest in each new generation.”
After Henry (above), we have Harry’s grandparents, Fleamont (who invented Sleekeazy’s Hair Potion, effectively quadrupling the Potter family fortune) and Euphemia, who were childless for many years until their son James Potter was born.
Fleamont and Euphemia lived long enough to see James marry a Muggle-born girl called Lily Evans, but not to meet their grandson, Harry. Dragon pox carried them off within days of each other, due to their advanced age, and James Potter then inherited Ignotus Peverell’s Invisibility Cloak.
For the entire article about the Potter family, check out the brand-new Pottermore.com.