The Davidian dynasty can be traced, generation by generation, for about 1,500 years, from Boaz and Ruth in the 11 century BCE all the way to the leaders of the Jewish community in Babylon in the fourth century CE.
How much of it can be scientifically demonstrated, I do not know. The existence of King David himself was proven by an archeological finding at Tel Dan just a few years ago. Here's a new story, in which Eilat Mazar, an archeologist digging in Jerusalem, says she has found a wall of the city that was constructed by Solomon.
The item notes that not all archeologists will welcome this interpretation of the finding, since some of them regard the Biblical stories of the Davidian kingdom as unreliable. It's an argument which has been raging in the academic journals for the past 20 years, perhaps more. My impression is that the conservatives, the ones who credit the Bible with greater reliability, are slowly gaining ground (no pun intended), but then again, that's also where my preferences would naturally lie, and I'm not reading all the academic stuff, so who knows.
Disagreeing about the House of David is not a new pastime. Just this morning I passed a section in the Sanhedrin tractate in which rabbis from the second and fourth centuries CE tired to agree on how it could have been that David wedded both Meirav and Michal, two daughters of Saul: wedding sisters is forbidden. (Yes, I've heard of Leah and Rachel, but that's a different story). One possibility is that Meirav died before Michal was wedded, another possibility is that there was a legal screw up in which Meirav ended up never wedding David, and the Gemarah hacks away at it for about a page. The axiom of the discussion is that these ancient Biblical figures were organizing their lives according to the very detailed rules being formulated in the Mishnaic era - an improbable assumption. Yet the attempt to look back 1,400 years and fit old events into a contemporary legal framework is ultimately no more silly than peering back 3,000 years so as to fit them into a contemporary political one, is it?