After a brilliant academic career in Rome, Billot was created a Cardinal by Pope St. Pius X during consistory held on 27 November 1911. He participated in the conclave of 1914, which chose Giacomo della Chiesa to walk in the shoes of the fisherman. Prior to Benedict XVI, della Chiesa was the most recent Pope Benedict. Billot also participated in the conclave convened in 1922 on the death of Pope Benedict XV, during which Milan's Achille Ratti was picked. It was Ratti who chose the name Pius XI.
From what I can glean, Billot's resignation, given to Pope Pius XI on 13 September 1927 was not forced, but chosen and tendered at the end of a stormy meeting with the pope. Pius XI formally accepted the resignation eight days later, on 21 September 1927. The Holy See did make Billot's resignation public until December 1927. The dust-up with the Holy See occurred as a result of then-Cardinal Billot's involvement with Action Français, a far right, French monarchist movement. It seems the immediate cause of the meeting that resulted in Billot's resignation was Pius XI putting the movement's self-named newspaper on the Holy See's Index Librorum Prohibitorum, or "Index of Forbidden Books."
Pope Pius XI was quite correct not to let the Church be co-opted by Action Français and to place the newspaper on the Index, which did not stop anyone from reading it, but made clear the Holy See disapproved of the movement. The movement was anti-Semitic and favorable towards fascism. This is why the leadership of the movement actively collaborated with France's infamous Vichy regime. It bears noting that Action Français certainly had less extreme members, but the movement as a whole was certainly beyond the pale. The movement advocated for the restoration of the French monarchy, the repeal of the 1905 law that separated the Church from the State, in addition to making the Roman Catholic Church, once again, the state religion.
After his resignation, Billot remained a priest and a Jesuit for the rest of his life. He died at age 85 in 1931.
There are eleven canons (349-359) in the current Code of Canon Law that deal specifically with Cardinals of the Holy Roman Church. While none of these deals specifically with a cardinal resigning, or being removed from the college, it seems to me as a non-expert that Billot's resignation, or, more specifically, Pope Pius XI's acceptance of it, shows that it is possible for a cardinal to resign as long as the Roman Pontiff accepts his resignation.
In addition Cardinal Law and now Cardinal Mahony, another modern Cardinal who brought disgrace on himself and the Church was the Austrian Benedictine Cardinal Archbishop of Vienna, Hans Hermann Groër. While Groër was removed from all ecclesiastical duties and responsibilities, he died a Cardinal.