It really depends on what you mean by "crazy" and who among Rome's chroniclers you serve to believe.

It's unequivocally clear from the historical record that Nero was debauched narcissist who didn't feel himself bound by the conventional morality of his day. This was was a man who, after all, traveled through the empire participating as the prime contestant in musical festivals which he ordered to be held; castrated a man (illegal within the empire at the time) "married" him and dressed him as the emperess, and made him his wife; and, after the Great Fire of Rome, had made a massive bronze statute of himself nearly as tall as the Statue of Liberty at the smallest.

What isn't so clear now is whether Nero was was simply perverse but also measured in his excesses and acting with genuine concern in his governance of the empire(as Pliny and Tacitus depicted him and how he is now generally viewed) , or completely without conscience and unhinged (as writers like Suetonius, Dio Cassius, and Sextus Aurelius Victor portray him).

For instance:

In the year 64 A.D., a devastating fire occurred in Rome. The Emperor, who was at Antium, on learning of the conflagration, returned to the city and undertook relief measures. He threw open his gardens and public buildings to the dispossessed populace and brought in grain from the neighboring towns. But, according to Tacitus, "his measures, popular as their character might be, failed of their effect; for the report had spread that, at the very moment when Rome was aflame, he had mounted his private stage, and, typifying the ills of the present by the calamities of the past, had sung the Destruction of Troy."

Suetonius, writing at almost the same time, tells the story as if it were a fact, changing the details as follows: "Viewing the conflagration from the tower of Maecenas . . . he sang the whole of the Sack of Ilion in his regular stage costume."

Reading Tacitus and Pliny we see a man who, loved his wife greatly and mourned her heavily after her death; personally led the relief of Rome after it's Great Fire and dedicated much time and effort to it's reconstruction afterwards;and took much interest in Rome's war with Parthia and with putting down of revolts (inc. one which resulted in the destruction of the Temple in Jerusalem). We see a man who, though perverse, was capable of feeling genuine love and felt a desire to govern well.

Compare this to the completeness of what Sextus Aurelius Victor says of him in the Epitome De Caesaribus:

Domitius Nero, scion of Domitius Ahenobarbus, his father, and Agrippina, his mother, ruled thirteen years. 2. For five years he seemed bearable, whence some have reported that Trajan was accustomed to say that the principes as a group were far different than Nero -- for a five-year period. 3. In the city, he constructed an amphitheater and baths. 4. Pontus he reduced to the status of a province with the permission of King Polemo, from whom Pontus Polemoniacus is named, as likewise the Cottian Alps from the dead King Cottius. 5. For indeed he passed the remainder of his life shamelessly, so that it is embarrassing to commemorate any of this. He went so far that, sparing neither his own decency nor that of others, at last, veiled in the manner of a bride-to-be, [140] with dowry paid, with everyone crowding together in festive fashion, he was married in the presence of the senate. Covered with the skin of a wild animal, he used to forage the sexual organs of either sex with his face. He even contaminated with defilement his own mother, whom he afterward killed. After their husbands were slain, he married Octavia and the Sabina with the cognomen Poppaea. 6. Then Galba, proconsul of Hispania, and Gaius Julius usurped imperium. 7. When Nero learned that Galba approached and that the senate had resolved that, according to ancestral custom, when his neck had been thrust into a yoke, he was to be beaten to death with rods, he, completely deserted, left the city in the middle of the night with Phaon, Epaphroditus and Neophytus, and the eunuch Sporus, whom once, after he had been castrated, he had tried to transform into a woman; and he pierced himself with a blow of his sword, with the impure eunuch about whom we spoke aiding his trembling hand while, since no one had been found earlier by whom he might be struck, he soberly exclaimed: "So, do I have neither friend nor foe? I lived shamelessly, let me die shamefully." 8. He perished in the thirty-second year of his life. The Persians had been so delighted with him that they sent legations furnishing materials for constructing a monument. 9. Otherwise the provinces as a whole and all Rome exulted in his death to such a degree that the urban masses donned the caps of freedmen and celebrated manumission, as if they had been delivered from a savage dominus.

TL/DR - Nero was "hold music festivals in his own honor and marry, as "empress" a man he had castrated crazy" but probably not "screw his own mother, stomp to death his pregnant wife and sing while Rome burns" crazy.