Even some Christian scholars find the claims of Tertullian...less than believable:
http://www.earlychristianwritings.com/actspilate.html
It seems at least one respected New Testament scholar is also quite dismissive of the Tertullian story, however unpleasant it might be.
http://www.earlychristianwritings.com/actspilate.html
Quote:F. F. Bruce writes (The New Testament Documents: Are They Reliable?):
...
Then Tertullian, the great jurist-theologian of Carthage, addressing his Defence of Christianity to the man authorities in the province of Africa about AD 197, says: 'Tiberius, in whose time the Christian name first made its appearance in the world, laid before the Senate tidings from Syria Palestina which had revealed to him the truth of the divinity there manifested, and supported the motion by his own vote to begin with. The Senate rejected it because it had not itself given its approval. Caesar held to his own opinion and threatened danger to the accusers of the Christians."
It would no doubt be pleasant if we could believe this story of Tertullian, which he manifestly believed to be true but a story so inherently improbable and inconsistent with what we know of Tiberius, related nearly 170 years after the event, does not commend itself to a historian's judgment.
...
It seems at least one respected New Testament scholar is also quite dismissive of the Tertullian story, however unpleasant it might be.