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....there are always gaps and breaks in the chain...But there is no actual standard of what gap of time is acceptable, and what gap would break succession. Thus, it is simply impossible to say what gap is acceptable. For example, according to a typical list of popes (example) there was no pope during the whole years 259, 305-307, 639, 1242, 1269-1270, 1293, 1315, and 1416, not to mention the many partial years. That's over a half dozen breaks of over a year.
....Being deposed?
....Being outrageously sinful?
....Being a heretic?
....Leaving Rome?
....Needing an Ecumenical Council to Jump-Start it?

....How much more broken could it really get?

Offering up the guise of apostolic succession raises all sorts of questions when Rome has to "self-correct" itself from bad popes. The unreliability of apostolic succession is demonstrated by Rome's having chosen "bad" popes in the first place, demonstrating that they failed to heed the infallible and inerrant guidance of their predecessors:

Protestants have reacted strongly against the doctrine of apostolic succession. They have done so in a number of ways, historical and theological. One of these ways is by affirming the apostolicity of the church. Apostolicity may be defined as receiving and obeying apostolic doctrine as it is set forth in the New Testament. In matters of doctrine and life, Protestants permit no ultimate appeal to traditions that are distinct from canonical Scripture.... ....Even if it were historically provable that there was an unbroken succession of bishops from the first century to the present day Roman Catholic bishops (and it is not), Protestants would still demur to claims of Roman authority based upon apostolic succession. It is the apostolicity of the church that counts. And it is precisely by the standard of apostolicity that the Roman Catholic Church is measured and found wanting.
-- from the thread Apostolic Succession and the Roman Catholic Church
The theory behind apostolic succession is that God's authority, to be meaningful and effective, must be embodied in men today who have the same kind of authority [as the original apostles]. But if you will read carefully the following passage, you will see that this is not true at all.
In 1 Corinthians 5 Paul - who was not physically present in Corinth - wrote to them to tell them what to do with respect to a discipline case. He said (in 5:4-5) "In the name of our Lord Jesus Christ, when you are gathered together, along with my spirit, with the power of our Lord Jesus Christ, deliver such a one to Satan for the destruction of the flesh, that his spirit may be saved in the day of the Lord Jesus." So you see, Paul did not pass on his authority to another man so that he could be there in Corinth. No, Paul said, in effect, if you will do what I as an apostle now instruct you to do then I will be with you in spirit, and you will also have the power of our Lord Jesus with you, to deliver that man to Satan etc.
So, to put it simply, the Reformers realized that there was no need for apostolic successors. No, the need was simply to have the apostles themselves with us through their inspired and inerrant teaching. And that is what we have in the New Testament.
The apostles never wrote anything that ever has needed or ever will need correction because they were inspired by God. Surely a person of average intelligence should be able to see that this has never been true of other men in history no matter how strongly they may have believed themselves to be apostolic successors!
-- from the thread Apostolic Succession and Protestantism