Short answer: Yes.
None of the Julio-Claudian emperors who followed Augustus (Tiberius, Caligula, Claudius, and then Nero) were particularly stable, but Nero has always been regarded as the worst of the lot. The debilitating dynastic decrepitude that afflicted the Julio-Claudian emperors was not particularly a direct hereditary problem -- they were either adopted or were off the main genetic path -- but there had already been centuries of inbreeding among all of the ancient Roman elite families. And the water they drank did arrive through lead pipes. Insanity caused by imbibing ionized lead is now easily documented, and the ancient Roman upper classes, who had indoor plumbing, drank more lead than the poor, who got their water from public fountains. In addition, upper class Romans were inveterate sippers, and, although they drank their wine well watered (2-3 parts water to one of wine), they always drank enough to keep up a perpetual buzz. Nero was particularly famous for his carousals.
Longer answer: Maybe yes, maybe no.
Nero's mother was Agrippina the Younger, a great-granddaughter of Augustus, and his father was Gnaeus Domitius Anenobarbus, scion of another of the great Roman senatorial families. Gnaeus died when Nero was a child, and Agrippina quickly remarried. The stepfather soon died under suspicious circumstances (poison was rumored), and Agrippina then became the incestuous wife of her uncle, the Emperor Claudius (who himself had just divorced and bumped off his own first wife, Valeria Messalina, due to Valeria's public promiscuity.)
Agrippina persuaded Claudius to favor Nero for the succession, over the rightful claim of his own son, Britannicus, and arranged for Octavia, the daughter of Claudius, to marry Nero. Nero succeeded to the throne in 54 AD when Claudius died suddenly after eating a plate of mushrooms supplied by Agrippina. Rumors of more poison were again rife, but no real evidence was ever brought forward (and the forest porcini mushrooms, favored then as now, do contain a drug that is heart-stopping poisonous if too many are eaten and if taken with the more than sufficient doses of alcohol that Claudius habitually drank.)
Nero was not yet 17 when he was proclaimed emperor, and, at the start, he was firmly under the thumbs of Messalina and her ally, Sextus Afrianus Burrus, the Praetorian Prefect, who had swayed the Praetorian Guard to raise Nero as Emperor.
Burrus and Nero's old tutor, the Stoic philosopher Seneca, owed their positions to Agrippina, but they soon tired of her arrogance and encouraged Nero to act independently. Nero's relationship with his mother quickly soured, and in 56 he forced Agrippina into retirement. From then until 62, Burrus and Seneca were the effective rulers of the empire.
Under the influence of Agrippina and later of Burrus and Seneca, Nero started out well. He ended the more extreme features of the later years of Claudius' reign including secret trials presided over by the emperor, he curbed the power of corrupt freedmen in the bureaucracy, and he returned some powers to the Senate. Contemporaries, during this period, described Nero as "a handsome young man of fine presence but with soft, weak features and an unsettled spirit", and, up to 59 AD, Nero biographers cited only his acts of generosity, sympathy, and mercy. He forbade contests in the circus involving bloodshed, banned capital punishment, reduced taxes, and allowed slaves to bring civil suits against unjust masters. Nero himself pardoned anti-imperial pamphleteers and even officials who actively plotted against him. Secret trials were rare, and treason laws lay dormant. Claudius had put 40 senators to death, but, between the senatorial murders instigated by Agrippina at the time of Nero's succession in 54 and the year 62, there were no similar anti-Senate moves. Nero encouraged the arts and inaugurated competitions in poetry, in drama, and in athletics as counter-attractions to gladiatorial combats. He provided aid to cities that had suffered disaster and, at the request of the Jewish historian Josephus, gave direct aid to the Jews -- and indirectly, through the jews, to their supposed sub-sect, the Christians.
Although Seneca's stoic philosophy was based on forbearance and self-denial, he and Burrus failed to stem Nero's rising lusts. Nero's nocturnal exploits, including murders and other street crime, were a scandal as early as 56, but Nero's real brutality became apparent in the 35-month period between the murder of his mother at his orders in 59 and his similar treatment of his wife Octavia in 62. Nero's growing dislike for his mother was fueled partially by her own plots to regain power, but, even more, it resulted from his desire to prove his independence. He ultimately lured her onto a specially built self-destructing boat, but she swam ashore when it collapsed at sea. He then had her strangled by her own military escorts. He had Octavia murdered to clear the way for a marriage to Poppaea Sabina, the young wife of the senator (and, post-Nero, a very-short-term emperor) Otho. Nero married Poppaea in 62, but she died in 65, and he again remarried.
Along the way, Nero realized that, whatever he did, he didn't have to fear retribution -- his ravenous appetites for blood, sex, food, and drink went unchecked. At the same time he began to act out his extraordinary artistic pretensions, performing publicly as a poet, musician, and actor. He drove in chariot races (sometimes incognito), but had the sense to avoid the gladiatorial games. He abandoned any pretence of following Seneca's stoicism and instead developed strange religious enthusiasms and sought out preachers in extravagant religious cults. Rumors spread (and were later written up by Roman historians, who were captivated by the succeeding Flavian dynasty) of bloody orgies in the palace.
And then there was the fire. Catastrophic fires were no rarity in Rome, but the one that started in a shop near the Circus Maximus on the morning of July 19, 64 AD, was exceptionally large, and it raged for nine days. There is no evidence at all that Nero had anything to do with setting or sustaining the fire, but Nero's reputation had sunk so low that it was easy for people to believe that he was the culprit. Nero was in his cool seaside villa at Antium (Anzio) 35 miles away, and, though he didn't fiddle while the city burned, he did later perform an Ode, said to have been written at the time of the fire, comparing the Roman conflagration to the destruction of Troy. Nero's rapid incorporation of much of the burned-out area into his planned gigantic Domus Aurea pleasure palace -- had it been completed, it would have encompassed a third of Rome -- did nothing to allay rumors of his responsibility.
According to the Annals of the Roman historian Tacitus and to the biography of Nero written by the Suetonius, Nero in tried to shift responsibility for the fire on the Christians, who were already becoming popularly identified as undesirables. Until this point, successive Roman governments had not clearly distinguished Christians from Jews, who, under Nero had protected status. By identifying the Christians as a separate and dangerous cult, Nero initiated the subsequent policy of halfhearted persecution of the Christians, and, in the process earned much of his disrepute among later (also captivated) historians in the Christian tradition: it was during this "first persecution" that Peter and Paul were reportedly martyred in Rome.
Needless to say, Nero made many enemies. There were military successes in Armenia, but the other provinces were in uproar, mainly because of new taxes levied to support Neronian extravagances. Gifts to his favourites alone were claimed to have amounted to more than 2,000,000,000 sesterces, a sum that was several times the annual cost of supporting the armies. A revolt in Britain was headed by Queen Boudicca (or perhaps Boadicea) in 60 or 61, and an insurrection by Zealots in Judaea lasted from 66 to 70 (finally put down by Titus.) Closer to home, the conspiracy to make Gaius Calpurnius Piso emperor in 65 AD revealed the extent of the discontent and the diversity of his adversaries. It included members of the senatorial and equestrian ranks, military officers -- a particularly ominous sign -- and philosophers. Advance warnings, perhaps from slaves informing against their masters, allowed Nero to spike the plot. Even in this instance, Nero did not altogether abandon his lenient attitude: of 41 participants in the Piso conspiracy, only 18 died (including his old tutor Seneca and the poet Lucan). The others were exiled or pardoned.
Shortly thereafter, Nero embarked on an extended visit to Greece where for 15 months he trod the boards of Greek theaters and let his hair grow out in the style of Greek ascetics. During this sojourn he freed several Greek cities in honor of their glorious history. Nero returned to Rome in February 68 and found the city firmly set against him. His enemies had previously included the senatorial and equestrian classes, and now his artistic, religious, and philosophical pretenses alienated the highly moralistic middle class of merchants and landed farmers who provided most of the lower rank of the military officer corps. Even the military rank and file were scandalized by his stage roles as "pregnant women and slaves about to be executed." Provincial governors, Galba in Spain and Julius Vindex in Gaul, openly revolted.
Finally, in 68 AD his palace Praetorians turned against him, and his bureaucracy fled by sea from Ostia. Nero had to run, narrowly escaping the ever volatile Roman mob. Finally cornered by a Praetorian chase squad just outside Rome, he committed suicide either by his own hand or with the assistance of two of his slaves.
The Praetorians and the mob soon regretted the chase as legions loyal to rival claimants to the imperial throne arrived to fight bloody battles inside Rome's walls. Vespasian, the smartest of the four contending "Emperors", sent in an army from Germany but prudently stayed at his headquarters in Alexandria while the other claimants killed each other off. Vespasian picked up the pieces and founded the Flavian dynasty, and that's another story.
Nero is all over the Internet and much "captivated" history is still being purveyed there, even by usually reputable web sources. The "primary" Roman and Christian sources, neither really contemporary nor unbiased, are all available in English:
Tacitus -- extract: Emperor slays mom: http://members.aol.com/zoticus/bathlib/nero.htm
Dio Cassius -- the fire: http://www.fordham.edu/halsall/ancient/diocassius-nero1.html
Suetonius -- Nero biography from The Lives of the Caesars: http://www.fordham.edu/halsall/ancient/suet-nero-rolfe.html
Tertullian (Christian) -- Apologeticum: http://www.tertullian.org/works/apologeticum.htm
Catholic Encyclopedia: http://www.newadvent.org/cathen/10752c.htm
Nero's Persecution: http://www.cptryon.org/compassion/sum00/martyrs2.html
Claudius and Nero from ThinkQuest: http://library.thinkquest.org/12654/julio2.html
Nero from about.com: http://ancienthistory.about.com/cs/nero/