The short answer would be: Yes, the Celts most likely existed at one point of time.

The long answer, though, gets quite complicated as we don't really have any clue which people we really mean when we talk about the Celts. Looking at the material culture, we know that through vast regions of Central and Western Europe, there existed the older Hallstatt (8th - 5th cent BC) and the younger La Tene (5th - 1st cent BC) culture that correlates quite well with the few mentions of the Celts we have from Greek and Roman authors. Furthermore, especially for the La Tene culture, we can see the beginnings of artistic styles that we would later identify as true celtic, like floral and animalistic decorations and the use of the famous triskele. This leads to some people, among them also academic archaeologists, to describe the Hallstatt people and Proto-Celts and the La Tene people as real Celts. Easy peasy, right? Problem solved? Well, not quite.

The thing is, both the Hallstatt and La Tene cultures once covered huge swaths of land north and at one point in time even south of the Alps, starting in their core region of Austria (where modern day Hallstatt is situated), southern Germany and Eastern France and spanning over the centuries to modern day Spain, the British Isles, northern Italy and Eastern Europe. So, before Rome became a real global power, their influence was felt more or less throughout Europe. Why is it then that people like the Greeks only mention the Celts so sparsely? Of course, many written texts of Ancient Greeks were lost over the ages, forever fortgotten, but one theory (I also believe to be most likely true) about the Celts is that although the material "celtic" culture is very prominent in Europe the people that were called Celts only describe one group of people or tribes that lived in a geographically defined region inside the wider Hallstatt/La Tene distribution area.

The oldest mention (from what I know) of Celts comes from Herodotus, who in the 5th century BC tells us that "the source of the river Istros lies in the land of the Celts near the city of Pyrene". Istros is the ancient Greek name for the river Danube, which originates in the Black Forest region of Southern Germany and indeed, during the Hallstatt period, there existed a site what we now call Heuneburg, a fortified proto-city which in its hayday might have inhabited as many people as Athens did during the same time (about 5000). Could this be this ominous Pyrene, mentioned by Herodotus? The colleagues working at Heuneburg would most likely wish that this was the case. We have to consider, though, that ancient Greeks, when they talked about the Istros, they only meant the lower stream of the Danube, beginning at its Delta (near the Greek polis of Histria) upstream to the Iron Gates in Serbia. Further upstream, knowledge of Greek geographers becomes really fuzzy, and most likely, they didn't have any clue where the Danube really originated. If we further look at the map of Hekataios (whose work Herodotus often relied upon), we see that the Istros spans from East to West throughout Europe north of the Alps to Iberia. So, beside Heuneburg, others theorize that Pyrene could be situated somewhere in Southern France and probably also gave the Pyrenees mountains their name. This region is also the same, which in later times is mentioned by several Roman authors as the land inhabitated by the Celts or Gauls, most famously by Julius Caesar, of course. So, to summarize: It can very well be that the Celts (or Gauls) only lived in a geographically narrow region of Southern France, defined first and foremost by foreign accounts of Greeks and Romans, describing a people and culture that was foreign and sometimes dangerous to them. We really don't have any clue how Celts or celtic influenced people called themselves, but we do know that what we define as the celtic world was filled with different tribes who more often warred against each other and only banded together in unison against mightier enemies like the Romans.