Well, this is a powderkeg that no one over the last 2,000 years have been able to answer. Why? Well:

Firstly, there are several stories about the establishment of Rome. Apart from Romulus and Remus there is the Story of Aeneas who was a hero from Troy. His story is the one that we get in Virgil's Aenied, however one must be careful when using this source as it is argubly a propaganda piece for Augustus, there is a scene where Aeneas has a shield that depicts the battle of Actium. Which happened 1,100 years later. There are others two, that I can go into detail if you wish.

There is also the issue surronding the overall story of Romulus and Remus, even writers such as Livy think is unlikely that the twins were saved by a wolf, Livy argues that the wolf is actually a prostitute (writing on my phone and don't have the quote I need to hand) due to similarity in the latin language. There is also the death of Remus, this would seem to show that fraticide was committed by the legendary founder of the city and therefore could be seen as almost acceptable. There is also the issue of the of the Rape of the Sabines. The establishment of Rome under Romulus is full of violence. Therefore there are many who change the story to make Romulus less of killer. For example one tale does not claim that Remus was killed and in fact lived to a ripe old age.

There was of course physical evidence of the life of Romulus. The Temple of Jupiter Stator that Romulus prayed in before battle with the Sabine men, the cave in which his brother and himself were cared for by the wolf. The house in which he lived on the Palatine hill. These remain way into the era of the Emperors so argubly there was some reason for these to be important, whether they really were indeed belonging to Romulus is not particualry easy to know.

The Name of Romulus also pretty much means Mr. Rome, the kings that came after him however did have proper names, Tarquinis Superbus, Severus Tullius etc. So this doesn't help his cause and the cause of the story.

So what do we almost know for certain using Archeological and literary sources:

• There was some sort of fortication built around the Palatine that dates back to 750-700 BC - This works for the idea of Remus jumping over Romulus' wall and then Romulus slaying his brother.
• Most writers agree that the early Romans were indeed asylum seekers, even Cicero appeals to this in some of his speeches.
• The Sabines and the Romans did fight. There is of course many reasons given why.

So did the story really happen. Very unlikey, was there a figure who united the various settlements around the hills of the Rome, possibly.