To answer, we have to understand "apostolic succession"; what it meant very early on and what it subsequently came to mean.
The early form of apostolic succession was the biblical idea that faithful men would teach the same to other faithful men.
Paul taught Timothy and Titus (among others directly and via the written word) and instructed us this way.
And the things that thou hast heard of me among many witnesses, the same commit thou to faithful men, who shall be able to teach others also. 2 Tim 2:2
Holding fast the faithful word as he hath been taught, that he may be able by sound doctrine both to exhort and to convince the gainsayers. Titus 1:9
This definition of apostolic succession is also found here.
Beloved, when I gave all diligence to write unto you of the common salvation, it was needful for me to write unto you, and exhort you that ye should earnestly contend for the faith which was once delivered unto the saints. Jude 1:3
The idea of the same doctrine being passed from elder to elder finds its way into the very early church fathers. As an example, here is Irenaeus regarding his mentor Polycarp.
- But Polycarp also was not only instructed by apostles, and conversed with many who had seen Christ, but was also, by apostles in Asia, appointed bishop of the Church in Smyrna, whom I also saw in my early youth, for he tarried [on earth] a very long time, and, when a very old man, gloriously and most nobly suffering martyrdom,3314 departed this life, having always taught the things which he had learned from the apostles, and which the Church has handed down, and which alone are true. To these things all the Asiatic Churches testify, as do also those men who have succeeded Polycarp down to the present time,—a man who was of much greater weight, and a more stedfast witness of truth, than Valentinus, and Marcion, and the rest of the heretics.
Some time later the idea of apostolic succession changed from elders who "teach the same" to one of sacerdotalism whereby the so-called succession became one of "holy orders" or a valid priest line (similar to a blood line).
This redefinition was done to accommodate large denominations such as Roman Catholic and Eastern Orthodox who do not teach the same as each other and thus as apostles, yet who want to claim that redefined apostolic succession. They just hope no one understands history and redefinitions.
So, the idea that Protestants only began in the 16th century is basically meaningless within the original definition of apostolic succession as "teach the same". Protestants were trying to reestablish the Christian idea of what that meant.
Extraneous dogmas developed apart from the Bible and much later than apostles were to be rejected by those who actually were of an apostolic succession. Teach the same as apostles as revealed in the Bible; when you do that you are of that apostolic line.