In short a provable link is impossible.

Let's say, for arguments sake, you want to prove you descended from King David.

You'd have to go back 3000 years in history through an unbroken line. David had children, and you have parents, so there's that. But you'd have to bridge a timeline that is fraught with wars, famines, upheaval, destruction, and entire civilizations crashing down.

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So what do you do?

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Genealogists take their cue from historians: the basis is sources, sources and sources. Any claim has to be backed up by source material to be valid. I will base myself on the European situation as I know that best.

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Let's look at the paternal line.

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In Western genealogy this means pouring over census records (modern era), civil records (which usually only started around the Napoleonic era), Church records (which usually are preserved from the sixteenth/seventeenth century onward), tax records (in some cases they go back to medieval times), legal records, etc. Every ancestor must be documented at some point (birth, marriage and death, composition of family, etc.) Not just because it's required, but also because every document holds clues about their lives, family and sometimes their crimes. Let us suppose you succeed in that. Chances are you will complete a genealogy of your ancestors going back to the 1600s, maybe even the 1500s.

Now it gets iffy.

Because your sources are church records, and if you are lucky, tax records. In every Western country nobody really bothered to keep them prior to the 1500s, with some very rare exceptions.

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If you are really lucky, you may find an aristocratic ancestor: aristocrats were obsessed with genealogy. There are several reasons given for this: claims of titles, prestige, ownership , family loyalties, but also prevention of marrying your own family (which was frowned upon by the Catholic church, with the notable exception of the Habsburgs). If there is a really important ancestor, like Charlemagne, this will be meticulously documented by the descendants as a source of pride. Even then, check and double check: I've run in to my fair share of "fantasy"-genealogies (I am looking at you, 18th century).

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Let us suppose that you manage this, now you've pushed back your genealogy to the medieval period. And then it basically stops. The records are either destroyed or never kept. But wait, there is that important aristocratic ancestor. You now have to rely on genealogies written down by monks relying on oral traditions: Pete begat William, begat Hank, begat.. etc. Serious documentation before roughly 1400 is scant or doesn't exist. Either thrown away, burnt during conquest or siege, or never written down.

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By the way, if the family fell on hard times and ended up as peasants, farmers, labourers or workmen: well, nobody really recorded those before the 1600s. You've got oodles of them running around, so why bother?

In other cultures - I did some preliminary research on South Africa for a Zulu friend of mine - records are scant or simply non-existent: non-European cultures didn't keep extensive records, but relied more on oral traditions. Unfortunately, oral traditions are hard to prove. So the further back you go, the less certain you are of the veracity of your claims. The oral tradition may jump over a generation or get to a dead end (the great-great-grandfather nobody wants to talk about), pool multiple generations into one person (example: three generations of "Solomons" may become one Solomon with an incredible long life. Yeah, that happens) or simply get mangled up because somebody forgot part or misunderstood it.

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So there is the problem with paternal lines.

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And the maternal line?

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A lot of cultures didn't deem women to be important enough to be recorded, unless they were really exceptional women. In medieval aristocracy in my neck of the woods (Holland) it has happened that if the woman was of higher birth than her husband, the children would adopt her last name. Usually, if you are lucky you might find a reference to a first name, but most of the time it is "anonymous".

So there is your maternal line problem.

Bridging a 500 years in genealogy is not that difficult if you can rely on Western sources.

Bridging a 1500 years in genealogy can only be done if you are related to (high) aristocracy.

Bridging 2000+ years in genealogy is impossible.

Now, I took King David as an example because there are people claiming to have descended from him solely on oral tradition. Even I can "prove" to have descended from him.

However, you can also take the genetic shortcut, check this article from Nature:

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Most Europeans share recent ancestors : Genetic sequences link far-flung populations and bear marks of historical events / Ewen Callaway

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So in a way, we all have the blood of every ancient religious founder through our veins. Provided they were born before that whole celibacy thing became a thing..