Trojan was the war, Trajan was the emperor. Most classicists do not classify any of the emperors as ALL good or bad, and in fact it has recently come into vogue to apologize for Nero to some extent and repair his reputation. Look up any biography on Nero in the last ten years and you'll find some musings about whether he has been unfairly treated.

Basically, Caligula, Nero, and Domitian all had difficult relationships with the Senate and were assassinated. Trajan and Hadrian were military men who not only had the backing of the Roman army, but also did not openly antagonize the Roman senate. The emperor is often portrayed as the greatest power, but the farther on you go in the Principate, you see more and more heirs are picked with the support of the army in mind.

So basically, it comes down to the stability of the empire. Caligula, Nero, and Domitian arguably created too much inner conflict within Rome to be "good" emperors. Trajan and Hadrian, with the support of the army and the Senate, were able to expand Rome's borders and focus their attention out towards the Provinces, gaining wealth for Rome.

I would challenge a little bit the presumption that they were all regarded as the good and bad emperors by the senate at the time. As with all history, the remaining sources are often fragmentary and biased in the first place making a black and white interpretation difficult.

Part of the problem is that as /u/Participlingdangle said, the relationship with the senate was pretty important to how they are represented in the ancient sources (as senators were the ones with the time and money to waste their day writing histories and political commentaries!).

But it's also important to remember that while Caligula was in power for all of 4 years, Nero was in power for 14 years, Domitian 14 years and Commodus (who also often gets the accolade of a 'bad' emperor... especially after Ridley Scott's Gladiator) was in power for 15 years.

The point I'm trying to make here is that the latter 3 remained in power for a significant amount of time, suggesting that (although I am guilty of generalizing quite a lot here too), they were to some extent able to rule the empire.

It is important to bear in mind that for instance Nero was under the 'good' influence of Seneca and imperial freedmen for the early part of his reign. etc. But it nonetheless suggests that the whole shebang is more complicated than it would first appear.

Some of the emperors have suffered part form the ancient sources and part from the later more modern treatment. For instance: Caligula in the X-rated film with Malcolm McDowell, Peter O'Toole, Helen Mirren et. al. Has become a character that we can 'love to hate.' Nero similarly in Quo Vadis has become a mockery of what it means to be a successful leader, especially in connection with burning cities.

But, they are not always depicted as bad (well apart from Caligula). For instance the ancient sources note that the fire and Nero's reaction was not all that it seemed:

"When the fire started, Nero was in Antium and did not return to the city until the fire was nearing the palace he built to connect the Palatine Hill to the Gardens of Maecenas. The fire, however, could not be stopped before it consumed the Palatine, the palace, and everything around it. By way of relief for the people made homeless by the fire, Nero opened up to them the Campus Martius, the monuments of Agrippa, and even his own gardens [on the Vatican Hill], and erected temporary structures to house the large numbers of homeless. Food supplies were brought upstream from Ostia and neighboring towns, and the price of grain was lowered to three sesterces a peck. But all of these efforts, although popular in nature, won Nero no favor with the people, since the rumor had surfaced that while the city was still on fire he got up on his private stage and sang his poem “The Fall of Troy,” noting the correspondences between the present calamities and that ancient catastrophe." Tacitus, Annals 15.38-41

after the burning of Rome, Nero starts a urban planning scheme in an attempt to make the city better and safer:

"Building heights were regulated and porticoes protected the fronts of apartment buildings. Nero promised that he would build the colonnades at his own expense, and that he would hand properties back over to their owners when he finished clearing away the rubble. In addition, he offered a reward, varying according to each person's station and the size of his property, for rebuilding homes and apartment buildings within a certain time. He also designated the marshes of Ostia as a land-fill for the rubble, which was taken down the Tiber on boats that had unloaded their shipment of grain. As for the new buildings themselves, Nero stipulated that a certain portion of them must be built without wooden beams, using either Gabine or Alban stone [= peperino tufas], since this type of stone stands up to fire.To increase the volume of the water-supply and deliver it to more places in public, assigned to stop the illegal tapping of the aqueducts, and each building (which could no longer share a wall with another building) had to have fire-fighting equipment on the premises." Tacitus, Annals 15.43; also see Suetonius, Nero 16.1

So a mixed picture has emerged, one in which the 'bad' at least are displaying traits that we would normally just associate with the 'good.'

The flip-side of the coin is that the good were not always 'good.' There is evidence to suggest that Hadrian was not universally liked and that it is our modern perspective that has granted him the accolade of a 'good' emperor. For example we have this source:

"[Trajan] advanced as far as the fringes of India and the Red Sea. He also created three provinces, Armenia, Assyria and Mesopotamia, including those tribes whose lands stretch as far as Media … [Hadrian], envious of Trajan’s military glory, immediately abandoned the three provinces that Trajan had added and recalled the armies from Assyria, Mesopotamia and Armenia, wanting the imperial frontier to be the Euphrates." Eutropius 8.3.2 + 8.6.2

Although this doesn't show that Hadrian was somehow bad, it does cast him in something of a of a different light. Hadrian's personal life was also something of a feature of gossip and speculation particularly his relationship with the young Antinous. That this was a homosexual relationship was made clear in the ancient sources (the Romans would not have found this remarkable, there is no word for homosexuality in Latin), but the shock is expressed in the manner in which Hadrian publicly mourned the sudden death of Antinous was unprecedented.

That's all I got time for now, but I hope I've managed to highlight how much the dichotomy between 'good' and 'bad' is really unhelpful in evaluating how the emperor actually was.

For a more balanced view of Nero (the good the bad and the ugly) check out the BBC drama documentary which had Mary Beard as the historical supervisor.

For a bit of fun and a catchy tune see the BBC comedy sketch about the bad emperors