The Medici family came from the Mugello region, north of Florence. Quite likely their family moved there for job opportunities and through sheer persistence and willpower ascended very well. Some of them got ennobled before the main line did. The main line did have aspirations to nobility and it took time for one of their relatives to finally get ennobled by the Holy Roman Emperor to become (Grand) Dukes of Tuscany.
(Like I said elsewhere, Italy’s historic tie to the Holy Roman Empire may’ve been analogous to what the semi-autonomous republics within Russia or what Macau and Hong Kong are to China.)
They also begat a woman named Catherine de Medecis who was accused of witchcraft by jealous folks. (She wasn’t infertile as she did beget children with her husband, if likely it’s her husband who’s got problems.) That and a pope challenged by Martin Luther who eventually created his own church instead. The Medici declined in power with one incompetent aristocrat.
However they still have descendants to this day, dealing with family history and stuff.