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Pope Pius XI took the cardinal's hat away from the great Jesuit theologian Louis Billot (1846-1931), because of the latter's associations with the conservative (albeit atheistic) French political movement Action Française. Cdl. Billot was the pope's top advisor, too. Yet, he remained a bishop.
Pope St. Pius X similarly removed that bishop's hat to show that he no longer has a right to exercise the office of a bishop publicly; he was suspended. It's not incorrect to address a bishop "Father", as bishops are have fullness of the sacrament of Holy Orders, but Pope St. Pius X obviously meant it as a humiliation for the prideful bishop.
In a dramatic ceremony at the Second Lateran Council, Pope Innocent II (r. 1130-43) deposed many bishops invalidly ordained by Antipope Anacletus II, saying (Doran & Smith p. 311):
Because the decrees of an irregularly appointed person [Antipope Anacletus II] are irregular, whatever he had established we destroy, whomever he had exalted we degrade, and however many he had consecrated we unordain and depose.
the pope called the creatures of the antipope forward by name and upbraided them ‘with indignation and reproach’. Next he ‘violently seized the pastoral staves from their hands, and shamefully pulled off the pontifical pallia, on which the highest dignity is based, from their shoulders, and also removed those rings by which betrothal to the church belonging to them is expressed, without regard for mercy’.
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Can a Catholic Bishop Lose His Ecclesiastical Rank without being Defrocked?
A bishop can if deamed appropriate by the pope be demoted to a lower ecclesiastical rank and can even be laicized if necessary.
Cardinals could lose their rank of cardinal, while still remaining a bishop. This is known as “losing their red hat”!
One clarification, I would like to bring up about cardinals loosing their red hats. Yes, cardinals can and have lost their red hat. Some cardinals have even voluntarily handed in their red hats back to the pope and some nominees, just like bishop nominees have refused these honours.
The ”ex-cardinal Louis Billot” in fact did not lose his red hat as implied by other answer, but he resigned from the rank of cardinal. His red hat was not taken away.
Cardinal Keith O'Brien, Also reigned being a cardinal.
In the Middle Ages, there existed a ceremony for taking away a cardinal’s red hat. The cardinal in question came before the pope bare foot with his his red hat on his head, knelt before the pope and then the Supreme Pontiff himself removed the red hat and after a lecture dismissed the individual in question. Back then priest could be cardinals also. I will try to locate this ceremony online as I believe it would be interesting here.
Now if priests can be demoted to being a deacon, it stands to reason, if circumstances merited it bishop could be demoted to functioning as priests only. Their episcopal faculties thus would be limited by the will of the Holy Father.
Over the centuries, I am sure this has happened.