I have always found the historical argument for the Catholic Church utterly compelling:
1. Jesus Christ established a Church (not several, or several thousand).
2. The only Church which can trace its lineage unbroken to the time of Christ and the apostles is the Catholic Church.
3. And therefore, Jesus Christ established the Catholic Church.
Why am I a Catholic? Because I wish to belong to the Church which Christ himself established, the Church of which he said, "the powers of death shall not prevail against it." (Matthew 16:18).
Some (few) Baptists have claimed that they too can trace their lineage back to the time of Christ and the apostles.
This idea was popularized in the early 20th century by Baptist pastor, and historian, James M. Carroll who wrote a book entitled Trail of Blood. In it Carroll claims that the Baptist church, as it is known today, descended through history under different names, such as the Anabaptists, Montanists, and Novations.
At first, this may sound tenable, but when you actually look at these groups, and what they taught, you see very quickly that their theology was anything but Baptist.
The Anabaptists denied that a person is saved by faith alone.
The Montanists taught that "God, not being able to save the world by Moses and the Prophets, took flesh of the Virgin Mary, and in Christ, His Son, preached and died for us. And because He could not accomplish the salvation of the world by this second method, the Holy Spirit descended upon Montanus, Prisca, and Maximilla, giving them the plenitude which St. Paul had not (1 Corinthians 13:9). 
The Novatians refused readmission to communion of baptized Christians who had denied their faith. 
"For proponents [of Baptist Successionism], writes Fr. Dwight Longenecker, "the fact that there is no historical proof for their theory simply shows how good the Catholic Church was at persecution and cover-up. Baptist Successionism can never be disproved because all that is required for their succession to be transmitted was a small group of faithful people somewhere at some time who kept the flame of the true faith alive. The authors of this "history" skim happily over the heretical beliefs of their supposed forefathers in the faith. It is sufficient that all these groups were opposed to, and persecuted by, the Catholics."
Thankfully intellectually honest Baptists, such as James McGoldrick who was once himself a believer in Baptist successionism are conceding that this "trail of blood" view is, frankly, bogus. McGoldrick writes:
Extensive graduate study and independent investigation of church history has, however, convinced [the author] that the view he once held so dear has not been, and cannot be, verified. On the contrary, surviving primary documents render the successionist view untenable. . . . Although free church groups in ancient and medieval times sometimes promoted doctrines and practices agreeable to modern Baptists, when judged by standards now acknowledged as baptistic, not one of them merits recognition as a Baptist church. Baptists arose in the 17th century in Holland and England. They are Protestants, heirs of the reformers. (Baptist Successionism: A Crucial Question in Baptist History , 1–2)
We should applaud these Baptists for desiring to be part of the Church Christ established, and then, with gentleness and reverence point them away from fallacious history to actual history, and let the evidence speak for itself. For as convert, John Henry Newman wrote: to be steeped in history, is to cease to be Protestant.