From the Historia Augustra regarding Hadrian, third of the "five good emperors," ruling in the early years of the 2nd century AD:

He held the quaestorship in the fourth consulship of Trajan and the first of Articuleius, and while holding this office he read a speech of the Emperor's to the senate and provoked a laugh by his somewhat provincial accent. He thereupon gave attention to the study of Latin until he attained the utmost proficiency and fluency.

Hadrian was Romano-Iberian, with his family having moved to the town of Italica near modern day Cordoba shortly after the Second Punic War. He would have been a fluent Latin speaker and would have been thoroughly Romanised in all respects (well, Hadrian had a bit of an obsession with all things Greek actually, but the point stands.) The empire was large however, and speaking fluent Latin doesn't mean that some haughty Italian senators aren't going to miss the chance to have a joke at a provincial, even if he's related to the current emperor.

As to other emperors, it's increasingly likely that they were accented as the number of "Italian" emperors declined from the 2nd century onward and provincial emperors became the norm. This may be particularity true of the so-called "barracks" emperors of the 3rd century who were proclaimed emperors at the behest of their legions and typically lacked the formal education that nobility throughout the empire would have received. For example, here's what Herodian has to say about Maximinus Thrax, the Thracian peasant declared emperor following the assasiantation of Alexander Severus:

His character was naturally barbaric, as his race was barbarian. He had inherited the brutal disposition of his countrymen, and he intended to make his imperial position secure by acts of cruelty, fearing that he would become an object of contempt to the Senate and the people, who might be more conscious of his lowly origin than impressed by the honor he had won. Everyone knew and spread the story that when he was a shepherd in the mountains of Thrace, he enlisted in a local auxiliary cohort because of his huge size and great strength, and by luck became the emperor of the Romans.

This doesn't sound like someone who'd have the well-manned speech of an Italian noble.

Sources:

Edward C. Echols - Herodian of Antioch's History of the Roman Empire

François Paschoud - Historia Augusta

A (hopefully sufficiently) related question: did a language barrier exist between eastern and western emperors, or the elites of the two halves of the empire?