Once again it comes down to who is telling the story. There are any number of anti-catholics who claim that there where two or three Popes at the same time. While the Catholic Church maintains that there can only be one Pope at a time. As a matter of fact there are some ten or twenty men who claim to be Pope today. Over the past 35 years there have been quite a few who made the claim. The fact remains even if more than one man claims to be Pope, there can only be one legitimate Pope.
Today if you do a web search for Pope Pius XIII, or Pope Michael I you will see two men who claim to be Pope, and claim that Popes John XXIII, Paul VI, John Paul I, John Paul II, and Pope Benedict XVI are all false Popes, that they are the only Pope. Well history, and common sense prove they are both wrong. They both have followers. They both have issued “Papal Proclamations” but that does not make them Popes.
The times that you (James) have mentioned, and that the source you posted speak of where times of great upheaval, however no matter how many claims that there where more than one Pope at any one time are made, it just can’t happen, any more than there can be more than one president of the United States. You can have one true president, and false claimants to the office. In England there is only one Monarch at a time, there can be pretenders to the throne, who claim to be Monarch.