It is widely assumed that the Carthaginian general Hannibal Barca crossed the Western Alps some 2200 Years ago with an army of men, horses and elephants in direction of Italy and the Roman Republic. But sadly, we have not found many evidence of that event and thus don't even know the exact route. What we also know is that Hannibal's elephants died during that route. It was more than 30 elephants and also we know that elephants are not really typical fauna of the Alps. That leads me to the question - how is it possible that we never found any elephant fossils in the Western Alps yet? Is it possible that the story of Hannibal crossing the Alps was made up? Did he just take the shortcut via Massalia and Liguria in the south? Did he use the Carthaginian fleet to invade northern Italy?
The simple answer is because bones are organic, and organic things don't last 2000 years. Even hard organic things like bones, except in very extreme (eg: rare) circumstance. Exposed bone, unless its somewhere with little life, will generally be gone within a year.
Usually when we talk of archeologists finding "bones" what they have really found is fossils of bones. These are not the bones themselves, but rock formed within the bone and replacing the organic matter as it dissipates so that it becomes more and more rock.
In other words, fossils are rocks in the shape of bones, not the bones themselves. However, in order for this to happen, the bone almost always has to get buried somehow.
Now picture an army moving over the alps to invade Rome, when an elephant dies. How likely is it that this army will order a halt so they can take the time and considerable effort to bury an elephant? More likely they'd just scavenge the carcass for all the meat they care to carry, and then march on. The rest of the carcass is of no use to them, so it would just be left there to rot. Within a year, if nothing comes along to bury any of it, it will be gone completely. Bones and all.
The absence of evidence is not proof, particularly when it comes to archeology where there's so much ground to [un]cover and so few people to do it, and in this case the proof is particularly difficult to find.
We don't know Hannibal's route across the Alps. The two Roman historians who are our primary contemporary sources, Polybius and Livy, were short on details and disagree on the route. So we don't know where to look, and the Alps present a lot of very difficult terrain to look through.
Then there's been 2200 years of time to bury or scatter any remains (but not sufficient to form fossils). It's not a matter of simply stumbling on them lying on the ground, this is enough time to bury cities multiple times over. Contemporary searchers have tried, and here's a Guardian article on their efforts. Note at the end how they have to dig 40cm into a bog before they reach the soil layer associated with Hannibal's crossing.
It requires deliberate effort, and a lot of money, to mount a search. Fortunately, modern archeologists don't have to dig to find likely spots, now they have ground penetrating radar, but even this is laborious and expensive. To get an idea of the labor involved, watch any episode of Time Team and see how little ground they can over in 3 days.
Neither Polybius or Livy, the 2 main accounts, mention how many elephants, if any, died during the crossing of the Alps. Appian's account says that Hannibal took 37 but also does not number those lost, if any, crossing the alps. Hannibal certainly had a number of elephants at the Battle of Trebbia, though all but 1 or 7 supposedly died in the cold weather afterwards.
Though elephants do not climb up mountains unnecessarily, they do so when it is part of their migration routes. Elephants can handle colder weather than they are used to, due to their great internal heat. Elephants are also very surefooted.
So it is perfectly possible that there never was even a single elephant bone or skeleton from Hannibal's army in the Alps to be discovered.
This thread in history reddit claims that someone found elephant poop, presumably in the Alps, and hopes to do genetic testing to discover which species of elephant they are from.