Of course one has to put into account the possibility that ancestries stretching back 2000 years back might not be correct. During the renaissance it was popular for monarchs to hire genealogists, who often constructed pedigrees going back to Odin and the kings of Babylonia. Descending from anitquity was also popular.
Having that said I find it possible, and perhaps even likely that there were some genuine connections between the new Elite of the Dark ages and the Roman one through marriage. It would seem unlikely that the Roman dynasties just disappeared. Proving it is another thing due to lack of sources.
It is also, as some have already pointed out often possible to back to for example Charlemagne if you find a connection to the nobility in perhaps the 16th or 17th centuries. And sometimes these connections come from surprising lines. I may have one of my own. My 3d great grandfather @Pål Björk was a shoemaker and former soldier in the cavalry. But it turned out that he had an ancestor @Ernst Laxmand von Rahden, who was a member of the German nobility and son of @Helle Laxmandsdatter Gyldenstierne. If I've understood it correctly she descended from Charlemagne in several ways.