There weren't really that many Caesars during the first century. Consider that Augustus was over 70 years old when he died in 14 AD, and Tiberius who succeeded him was already middle aged by then, meaning that he too was old when he died in 37 AD. I don't know if you've seen the graph that's been posted on reddit over the deaths of Roman emperors? I won't post it here since I don't want to encourage spreading it, but either way, being a Caesar wasn't truly dangerous business until after Pertinax.

Of the Julio-Claudian emperors, we only know that Caligula was assassinated. Nero comitted suicide, while Augustus, Tiberius and Claudius (despite rumours in eg. Suetonius) most likely all died natural deaths. During the year of 4 emperors one might argue that it wasn't ideal to become emperor, but looking at the situation realistically, it is no wonder that neither of Galba, Otho or Vitellus lasted more than a few months each. Vespasian, who came out on top in the end, was a respected general and well liked within certain parts of the empire, making his assumption of power a lot smoother.

And looking at the Flavian emperors then, Vespasian was already quite old when he came to power and he died a natural death; and so did his son Titus (albeit of an illness at a comparatively young age). Domitian was admittedly assassinated, but that was after ruling the empire horrendously for 15 years, making his demise less because of his position and more because of his decisions.

We're now at 96 AD, when Nerva becomes emperor. However, Nerva was chosen partly because of his old age, simply as a good replacement emperor after Domitians assassination, to give the senate time to find a fitting emperor. That person was Trajan, who ruled as emperor for 19 years before dying of natural causes. He was then followed by Hadrian, Antoninus Pius, Marcus Aurelius and Lucius Verus who all died of natural causes. Commodus, like Domitian, was assassinated for his actions rather than for his position.

So, arriving at Pertinax then, what changed in the empire that made the position a dangerous one? Well, Pertinax decided that it would be a good idea to buy the imperial seat from the Praetorian guard. But by doing so, he laid the foundation for an era we call "The Soldier Emperors". Interestingly enough, the first emperor of that era, Septimus Severus, was actually a good and popular emperor who reigned for many years. However, when he gave the empire to his two sons, Caracalla and Geta, after his death, things turned ugly. The two brothers could not get along and Caracalla subsequently ordered his brother assassinated. That was the start of a cruel reign, which eventually ended in the assassination of Caracalla.

Caracalla's reign was followed by roughly 30 years (217-245 AD) of assassinations and battles between aspiring emperors, during which the empire had some 20 different emperors. This would be the tipping point you're asking for. During this brief period, the empire was essentially ruled by the praetorian guard; and if the guard didn't like the sitting emperor, they replaced him. Meaning that while the position still was prestigious, it was extremely dangerous to have and essentially a death sentence, but a lucrative one. The pull for becoming emperor during these times would thus have been the money and power the position gave along with it being seemingly "easily accessible" for the right man (ie. a man who controlled an army). Diocletian's reforms changed things and the empire(-s) under him became more stable.