Probably not, given that the fiddle was not invented until a thousand years later. The historian Suetonius have Nero watch the fire from his balcony while reciting The Sack of Troy (a now lost epic) in his stage costume. But this is most likely a myth which Suetonius presents as fact. Tacitus, regarded a more reliable historian, states that Nero was not even in Rome at the time of the fire, but returned immediately when he heard the news and organized relief efforts. This sounds much more likely, since Nero was not actually that crazy. Tacitus also mentions that rumors existed that Nero instigated the fire himself and persecuted Christians to deflect the blame. But there is no indications that those rumors have any basis in fact.

Nero did like performing and actually toured the empire as an actor/singer/musician (playing the lyre, not fiddle!). This was considered extremely inappropriate by the upper class, since acting was regarded as a dishonorable profession on the level of prostitution and gladiatorial combat. Something fit for slaves and perhaps freedmen but certainly not for people of good family. There was even laws in place that free men which partook in acting (or prostitution or gladiatorial combat) had to give up certain civil rights. For the emperor to tour as an actor when he was supposed to win wars and rule the empire was deeply embarrassing to the senate. (The later emperor Commodus broke a similar taboo when he partook in gladiatorial combat.)

So the image of Nero reciting (and in later versions of the myth, playing the lyre) while watching Rome burn, was a powerful image of how they viewed Nero: An embarrassment totally lacking the virtue and determination expected by a roman public figure. But as much as the upper class despised Nero, it seems he was quite popular among the people, to the extend that when he was deposed and his likeness was stricken from public monuments, anonymous activists kept placing images of him around the city! The people even had a hard time accepting that he was dead, and in the yeas after there were a belief that he would return, and there were Nero "sightings" around the empire. One Nero impostor even caused a diplomatic crisis when he appeared in the Parthian empire years later.

In spirit, it's a lot more justifiable on the evidence than many people would pretend, though it probably isn't true. (In the letter it's of course trivially untrue, since there were no fiddles.)

Tacitus Annals 15.39, Suetonius Nero 38, and Dio Cassius 62.18.1 all report that Nero performed a song about the sack of Troy, accompanying himself with a lyre, during the fire. Here's Dio's lurid account:

While everyone was in this state and many, crazed by the disaster, were even jumping into the flames, Nero ascended to the roof of the palace, from which there was the best view of the fire, and putting on a lyre-player's costume, he sang the Sack of Ilion -- as he put it; though it looked more like the sack of Rome.

In Suetonius, the top of the tower of Maecenas is the setting; in Tacitus, it's a private venue. So the story is potentially justifiable by these accounts. So in spirit, the answer is sort of yes: the story is certainly strongly supported by the ancient sources.

The reason for caution is that from the point of view of a more determined effort to recover the reality, there's good reason to doubt all of these accounts. Suetonius and Dio are not reliable writers, in the first place. Suetonius was basically a gossip columnist, determined to cast the worst possible light on anyone he disliked, and Dio was happy to accept the word of gossips. They both insist that Nero himself started the fire; in Dio's case his purpose was to destroy Rome purely for the sake of destruction. That's not a reasonable way of interpreting the actions of any ruler. Tacitus is more reliable, and he's also more cautious: the way he puts it is that there was a report of Nero staging a performance of the Sack of Ilion.

Really what it looks like is that Nero got really, really unlucky in the PR game. He was known for performing lyre concerts; it's perhaps even possible that he staged a concert to commemorate the fire, or that he had performed the Sack of Ilion previously and that this was recalled at the time of the fire. He used the occasion of the fire to claim an enormous amount of land in the middle of the city for a very large imperial palace, and that allowed ancient conspiracy theorists to suppose that he wanted the fire to happen. According to Tacitus he was out of town at the time the fire started, but returned when the fire was approaching his house. Basically it was a perfect storm of bad PR: by choosing facts selectively, it was perfectly possible for enemies to concoct a story where he himself started the fire, where he had a well-known motive for doing so, and where he revelled in the destruction he had supposedly caused by performing a lyre concert, something that he was known for.

In his defence: it's just silly to suppose that he was responsible for the fire (and, as Tacitus points out, he wasn't in town when it started); and the fact that we can see how the story may have come about allows us to reject the ridiculous picture that Suetonius and Dio report of a man who just wants to watch the world burn. On the other hand, it's certainly legitimate to claim that the story that "Nero fiddled (or rather played the lyre) while Rome burned" is an ancient one, and one that emerged very shortly after the event.