I'm ignoring the very early church because it seems like that would bring up a lot of uncertainties, but is it likely, guaranteed, or very dubious that all bishops descend from a long line of bishops? One act of infidelity might confuse an entire line of kings, but consecration seems a little more documented and hard to fake. Are there any obvious kinks or false fonts that have tainted entire lines of bishops? I can't find much discussion one way or another, but I see a lot of antipopes, legal-but-not-valid affairs, and disagreements that might raise questions about unbroken succession.
I assume some of this is going to seem like speculation because there will be differences in viewpoint over valid consecration, but I don't really have a jumping off point and don't know about any questionable or controversial consecrations.
The original Patristic notion of apostolic succession, which gained ground from the late 1st century onwards, saw the bishops as successors to the Apostles because the taught the faith of the Apostles. This is seen in the writings of Ignatius of Antioch, Polycarp and the first letter of Clement of Rome. It is not at all clear when the office of bishops and presbyters became distinct, but it was certainly the common pattern of Christian organisation by the end of the 2nd Century. By then, lists of bishops were already being drawn up, the first extant by Hegesippus (c110-180) in Palestine. Thereafter, episcopal lists became very common, in, for example, the 4th Century Eusebius of Caesarea and the 6th Century Gregory of Tours, and pretty much every medieval chronicler one cares to think about.
Given the care that was taken over such matters (as succession from the Apostles was held to guarantee the apostolicity, and therefore the truth, of the faith taught), I would think that unbroken apostolic succession is very likely. That is not, of course, to say that any particular bishop has 'valid' orders - the criteria for validity in any case being somewhat different in the Catholic and Orthodox churches.