I remember looking into this at some point (I was a classics major who then got a history MA focusing on Medieval and Early Modern Europe, so it kind of went with the territory). I think I recall that there is a claim that the British royal family can trace lineage back to a specific Roman senator, and I'm sure various noble families make comparable claims, but at the end of the day there simply aren't any reliable supports for the claims, and they are exactly the sort of pseudohistorical claims a scholarly courtier creating a genealogy to flatter some noble or monarch would have embellished or invented, and exactly the sort that they almost could not possibly have found reliable documentary evidence for.
What I can say, is that the Roman nobility, to which most of the most famous Romans belonged, disproportionately died out. It was a society that had wealth qualifications for nobility, looked down upon making money by any means other than inherited wealth, and divided estates equally amongst children. Thus noble families seem to have typically tried to confine themselves to one or two children, lest the estate be divided up too far to maintain noble status, but in a pre-modern society where people often die unexpectedly and medicine can do little for infertility it's only a matter of so many generations until that strategy reaches a dead-end.
Practically everyone with an ounce of European blood probably has some ancient Roman ancestors at this point, but disproportionately few famous Romans have living descendants.
