Sort of, but not exactly. It's a good question that we can pretty much apply to any historical figure or event, but as for Alexander:

Take for example the Anabasis Alexandri written around the 2nd century AD, over 500 years after his death. It was written by the historian Arrian, who based his work on the primary contemporary sources of Alexander's general Ptolemy, Aristobulus of Cassandreia - a historian who accompanied Alexander, and Nearchos - another of Alexander's officers. In fact, when looking at the oldest surviving sources of Alexander's life and conquests, we find references to multiple different contemporary sources that have since been lost. While those ancient sources have been lost - they were still around and available to Arrian and Plutarch and others who wrote about Alexander. So while today in 2019, we can't for example verify that Ptolemy really wrote whatever Arrian attributed to him, scholars at the time would have been able to - and without getting too off topic - ancient Greeks and Romans had an interest in textual criticism and taking liberties with the source material would have been noticed by other scholars.

In a time when copying documents was done by hand, ancient libraries had scribes who would ensure that texts were copied accurately and what needed to be preserved. Copying a document took time and money - so for example, an ancient scribe might have decided that Arrian accurately gave us all the important information from Ptolemy and Aristobulus and chose to preserve Arrian's work as opposed to the original sources. Or maybe the original works became too difficult to translate centuries down the road. Or maybe they were lost in a fire. Or maybe we'll find a copy some day that's been hiding in some medieval library. These are the sorts of things that drive the study of history.

For other things to verify that Alexander existed, we have some things like the "decree of Phillipi" - an inscription found in Greece about a land dispute that mentions Alexander. There's a handful of other inscriptions dated to the proper time period found around Greece that make mention of King Alexander. Of course we also have non-Greek/Roman sources like the Babylonian chronicles - cuneiform tablets currently in the British museum that discuss about Alexander's victory over Darius and his entry into Babylon.

And then we have the physical record. We know cities were named for Alexander reaching far east into modern day Afghanistan and Pakistan that verify the written history of Alexander's conquests. Archeological records in these areas show us that Greek/Hellenistic people moved there. For example - Alexandria Eschate in modern day Tajikstan is thought to be the city referenced in ancient Chinese texts as Dayuan (Da, meaning "great" and "yuan" meaning Ionian). And we know the aftermath of Alexander's empire - such as the extent of the Ptolemaic and Seleucid empires that took over the far reaches of Alexander's empire. We know that ancient Indian sources refer extensively to the Greek people (they called them Yona - again coming from Ionian) living in the areas Alexander conquered and differentiated between them and the Persians.

So without the original sources - yes, it sort of is like a game of telephone because we have to trust that the secondary source got it right. But we also know that they were written in a time by historians who knew those primary sources could be referenced. We can see multiple secondary sources referencing the same information from lost primary sources as a sort of assurance about the accuracy of what was being copied from the primary sources. We have non Greek/Roman sources referencing Alexander. And we have the physical and architectural record that corroborates what the written sources tell us about Alexander's conquests and we know Greek people were moving in and in control of the areas Alexander is said to have conquered.