Even assuming that Rome had a number of Christians besides those mentioned in Paul’s letter to the group there, by AD 64, the entire city is unlikely to have had more than 200. (The whole world likely had no more than 1,500 or so; the Book of Acts and Paul’s letters paint a success story that just ain’t so.)
On a fairly conservative, but not extremely conservative estimate, Rome had a population of 500,000. Two hundred people, given the lack of electronic communications, newspapers, etc. would stand out even less in a city of 500,000 than they do today.
Now, the Roman historian Tacitus does mention a disturbance about a certain Χρήστος, but several observations need to be made. First, this is NOT the same as the Greek Χριςτός. The first Greek word means “excellent” or similar. It is nowhere used in the Greek New Testament, nor the Septuagint, the Greek translation of the Hebrew Bible. Instead, it was an epithet used about Greek Olympian divinities in general and Apollo in general. (Sidebar: nearly 200 years, Constantine would have a sun-struck vision that, had he not had a Christian mother, he probably instead would have interpreted as a vision of Apollo[or perhaps Mithras] instead; his first post-conversion coinage shows a beardless, Apollonian Jesus.)
Let’s say that Tacitus was unfamiliar enough with Judaism and with this nuance of Greek vocabulary that he used the wrong word. That, then, would mean he wasn’t familiar with Judaism in general, nor disputes among Jewish “sects.” He may have known enough to be familiar with the basic concept of the Messiah, the Christ, and that was, and thought Χρήστος was the right word.
In other words, he could be talking about a general dispute between Jews about the Messiah, and not about Yeshua bar-Yusuf at all. Second, he places this disturbance in the imperium of Claudius, not Nero. So, it could in no way be connected to any disturbance that caught Nero’s eye.
Between the likely generic Jewish dispute of Tacitus’ account and the small number of Christians at the time of Nero, and assuming Christians took to heart Paul’s “submit to the governing authorities” command of Romans 13, there seems to be no reason Christians were blamed for the fires.
Ergo, perhaps they weren’t blamed. Detailed accounts of this alleged persecution are found only in Christian writings, and not in any early ones. Clement of Rome nowhere mentions them in his epistle to Christians in Corinth, though it would have been an excellent example of steadfastness to pass on.
Update, May 26, 2024: A person at r/AcademicBiblical, in chat with me about a main comment of hers there, where she claims that 1 Clement, stripped of legendary accretions, is about Peter and Paul being killed by fellow Christians, nonetheless, in response to me, claims that the Tacitus is not an interpolation.
I offered as a sidebar the Option B, that, if genuine by Tacitus, it's still a double-bankshot smear, projecting back from his own interactions with Christians, of what was a non-event, so that he could smear Nero even more than Christians.
As for said person's claims rejecting the interpolation idea? They note that the Correspondence of Paul and Seneca mentions the Great Fire earlier. Two counterpoints: First, on a late dating, only about 20 years earlier. So, it's in the same milieu, even if earlier. Second, the 11th letter of said correspondence is generally believed to be by a second forger than most other letters. Severus himself?
Said person does reject an actual persecution, so they're solid there.
So, perhaps there was no Neronian persecution of Christians, even one just limited to Rome.
