Bishops ordain Bishops ...
According to Catholic teaching, only bishops can ordain new bishops — this is how the apostolic succession is passed-on from generation to generation. If there are examples of invalid bishops ordaining other bishops, these new bishops are not valid bishops either and the chain of apostolic succession is broken.
I have never heard the teaching magisteriumof the Catholic Church address this topic, rather, they seem to merely assume that apostolic succession was never broken. Evidence of this is never provided, nor are explanations given to explain why apostolic succession was not broken in cases where it appears to have been broken.
I provide examples of historical incidents when the chain of apostolic succession was broken. The teaching magisterium should address this topic (but I have no reason to believe that their answer would be satisfying in the slightest).
In addition, we should strongly suspect that there are many examples throughout church history in which bishops were not ordained by other valid bishops. So many times bishops were appointed by secular rulers. I have a hard time believing that in every case this was followed up with an ordination conducted by other validly-ordained bishops — this is never mentioned, but is merely assumed as having occurred.
I find it odd that the Catholic Church is so detail-minded in ensuring that everything is proper in allowing Catholics to get married, yet so seemingly disinterested in ensuring (and publishing) that every bishop was ordained by other validly-ordained bishops. I would expect these ordinations to be heavily documented but such does not appear to be the case. We should do to the church what she does to us: assume someone's credentials are invalid unless proved otherwise.
And even worse than improperly ordained bishops is the issue of unqualified bishops; of bishops who are immoral. We should assume that these "bad" bishops cannot validly ordain other bishops because they are not valid bishops themselves. It is unthinkable that Jesus would establish a church which depends on "bad" bishops to select and ordain other bishops. The New Testament and early church fathers emphasize repeatedly that church leaders are to be qualified. Why do we even have "bad" bishops and priests anyway? Shouldn't the Catholic Church get rid of them all as their first priority?
Pope John XII reigned 955-64.
He was corrupt, violent, and immoral ...
A coarse, immoral man, whose life was such that the Lateran was spoken of as a brothel, and the moral corruption in Rome became the subject of general odium.
A synod composed of fifty Italian and German bishops was convened . . . . John was accused of sacrilege, simony, perjury, murder, adultery, and incest
John XII re-entered Rome, and took bloody vengeance on the leaders of the opposite party. Cardinal-Deacon John had his right hand struck off, Bishop Otgar of Speyer was scourged, a high palatine official lost nose and ears.
He excommunicated all the bishops who could have elected another pope, thereby breaking the chain of succession ...
Refusing to recognize the synod, John pronounced sentence of excommunication against all participators in the assembly [the fifty mentioned above], should they elect in his stead another pope.
[In] 964, John held a synod in St. Peter's in which the decrees of the synod of 6 November were repealed; Leo VIII and all who had elected him were excommunicated; his ordination was pronounced invalid
John XII at once returned to the city, summoned a council, condemned Leo "one of the employees of our curia, who has broken his faith with us", and degraded those clerics who had been ordained by him [Leo, who was not a valid pope].
Once Pope John XII died, these excommunicated bishops chose a new pope. They seem to assume that they were no longer excommunicated once the pope who excommunicated them died. But they can't revoke their excommunication themselves, only a pope can do that. But there was no way to elect a new pope because all the bishops had been excommunicated. The chain of succession is permanently broken.
When the Emperor Otho I illegally brought about the deposition of the unworthy Pope John XII (Nov., 963), he equally illegally caused to be elected, to fill his place, a layman, Leo.
Soon after this John died (14 May, 964), and the Romans unwisely elected to succeed him the Cardinal-Deacon Benedict.
Pope Leo VIII was not elected properly and was, in fact, an anti-pope. An anti-pope does not automatically become the true pope just because the true pope dies.
If it be the fact, as is asserted by a contemporary, that Benedict acquiesced in his deposition, and if, as seems certain, no further protest was made against Leo's position, he may well be regarded as a true pope from July, 964, to his death in 965, about the month of March.
There was a long period of time in which secular rulers appointed bishops, often the second-oldest sons of other family members. It is unlikely that these new "bishops" were ordained by other validly-ordained bishops. Some quotations:
Younger sons of nobles were intruded into bishoprics, at times even into the papacy. Secular princes claimed lay investiture of spiritual offices.
New Advent Encyclopedia
European monarchies over control of appointments, or investitures, of church officials such as bishops and abbots.
It seems to me that once there are invalid bishops on the scene acting as true bishops (and ordaining other false bishops), it won't take long before there are no validly-ordained bishops left. Chaos and anarchy have a preference for creating disharmony, not harmony. In order to preserve the purity of apostolic succession (and guarantee it), it would be necessary to strictly follow the exact procedure of valid bishops ordaining new bishops every time.
St. Hilary of Poitiers was expelled from what is now France by a council of Arian bishops. Certainly ordinations performed by these heretical bishops were invalid — how can ordinations by heretics possibly be valid? Thus, the chain of apostolic succession was broken in France.
The Church was then greatly disturbed by internal discords, the authority of the popes not being so powerful in practice as either to prevent or to stop them. Arianism had made frightful ravages in various regions and threatened to invade Gaul, where it already had numerous partisans more or less secretly affiliated with it. Saturninus, Bishop of Arles, the most active of the latter, being exposed by Hilary, convened and presided over a council at Beziers in 356 with the intention of justifying himself, or rather of establishing his false doctrine. Here the Bishop of Poitiers courageously presented himself to defend orthodoxy, but the council, composed for the most part of Arians, refused to hear him, and being shortly afterwards denounced to the Emperor Constantius, the protector of Arianism, he was at his command transported to the distant coasts of Phrygia.
The Arian heresy was so pervasive for such a long time period and over such a wide area of the church that it is difficult to imagine how apostolic succession was maintained during this time. Certainly there were orthodox bishops but without records of who ordained who it is uncertain which ordinations were valid and which were invalid.
It was concerning this last council (359) that St. Jerome wrote, "the whole world groaned and marvelled to find itself Arian".