Gibbons "If a man were called upon to fix the period in the history of the world during which the condition of the human race was most happy and prosperous, he would without hesitation name that which elapsed from the accession of Nerva to the death of Marcus Aurelius." That comes to a period of 84 years, from 96 A.D. to 180 A.D. "Their united reigns are possibly the only period in history in which the happiness of a great people was the sole object of government."
Were these five people competent rulers or did they just happen to come to power when Rome was on the up and up?
You may be interested in this thread from a couple of days ago. The question there was "Why does Edward Gibbon consider 2nd century Rome to be the best era in human existence?" which is slightly different from your question, but the answers there are still interesting I think.
'The Four Good Emperors' might be a better term. While Nerva was a suitable stopgap between the Flavian dynasty and Trajan, he served for less than 2 years and really didnt accompished as much as the other 4. Nerva did however chose Trajan as his heir, which was pretty much the best decision possible for Rome in general, so for that alone, Nerva is 'Good'. As for the rest, they certainly did not come to power while Rome was on the rise. Trajan faced a Dacian menace that had been allowed to grow under the rule of Domitian. Not only did Trajan find victory in Dacia, he also conqured much of the Parthean empire in the east, a feat which had alluded both Mark Antony and Crassus. Hadrian, Trajan's succsessor, withdrew from Parthea, as he could forsee issues it would cause if it remained a direct povince. Instead he placed a puppet king on the throne and was able to effectivly deter Parthean incursions into Roman lands. The rest of the good emerors continued on Trajan and Hadrian's numerous public works and buildings, and archetecture flourished during these years. The emerors also oversaw a heathy economy, largly in part to the Dacian gold mines that Trajan had captured. So in short, the 5 good emperors really were good, and considering what had come before them (Domitian's practical but autocratic reign), and what would follow (The beginning of the end, starting with Commodus), their reigns can be seen as a breif Golden Period for the Roman Empire
In short, Nerva's predecessor, Domitian, had a poor reputation for a very long time given that the Senate by and large hated him. Thus, Tacitus, Pliny the Younger, and Suetonious write of Domitian as though he were a cruel and paranoid tyrant. His reputation is disputed by modern authors as recent numismatic and archaeological findings have provided a counterpoint to the traditional perception based on the above ancient writers, particularly by Brian Jones The Emperor Domitian. The revision of his legacy began in 1930 with Ronald Syme The Imperial Finances under Domitian, Nerva and Trajan who argued that a complete revision would be necessary due to those findings.
The Emperor coming after Marcus Aurelius was Commodus, who is still widely regarded to be rather off his rocker. Given that Gibbon is writing before these revisionist arguments are being made, it is easy to see from his perspective that there were '5 good emperors'.