I keep hearing this same story over and over in my readings of military history. Someone confidently proclaims a certain piece of terrain impassible, and that therefore they have no need to guard against the enemy army coming from that direction. Enemy army does so anyway. So why do commanders keep making this mistake throughout history?
Because usually the generals are right. History only takes note of the times the they’re spectacularly wrong.
For example, in the run up to every war that the French have fought with the Germans the French generals have correctly guessed that the Germans will not attack by way of the Swiss Alps and have thus always left their border with Switzerland relatively undefended.
And there are plenty of examples of generals who decided to ignore the experts who said such-and-such terrain was impassible, and tried to move their armies through said impassible terrain, and proved the experts were right–their armies were immobilized and cut to pieces.
I don’t doubt it, would just like to read about it, could you give us a couple examples? 
The Roman invasion of Germany in 9AD.
Hardly impassable, nor is it necessarily suicidal or crazy, but quite a lot of attempts to invade Russia throughout its history have ended with the winter being the ultimate harbringer of doom for the invading force. Into this list you can arguably include Battle of the Ice, The Great Northern War (or particularly one really bad winter), Napoleon’s Invasion of Russia, World War II and probably many more.
Hannibal coming over the Alps cost him a substantial number of his men and elephants. He also had to leave behind large chunks of his force to hold newly conquered territory in Iberia that he wouldn’t had to have if he’d just landed in Sicily or southern Italy. All in all he left 11,000 in new Iberia, 12,000 in southern Gaul, and lost maybe 15,000 crossing the Alps; so his initial force of 80,000 was down to about 30,000 once he got to Italy (11,000 Iberians left his force rather than leave their homelands but that loss is not due to the terrain issue mentioned in the OP).
While he was able to wreak havoc in Italy once he got there, he never had the men to hold any ground, nor to even attempt to seige the city of Rome when it became clear that was the only way he was going to accomplish his goals.
harbinger of doom that is
Interestingly enough, even Russia was out-wintered by Finland during the Winter War. The Russians were unable to take Finland, their troops demoralized and machines paralyzed by the weather, and suffered huge losses. The native Finns knew the terrain and climate and used this to their considerable advantage. It was Russia’s Vietnam.
Thus the moral of the story is - invade the south, starve the north.
Actually, no. Finland surrendered eventually, and had to yield over some considerable land concessions.
Afghanistan was USSR’s “Vietnam”, also due to some terrain effects on military ops, I think.
The Teutoburg Forest was considered impassible in the sense of the OP ala the alps and Hannibal? I’d agree it was a dangerous trek for the Romans, but not because the geography itself was going to kill them. Hannibal’s odyssey came to mind immediately when I asked for examples, but I can’t think of any more. Invasions of Russia, as noted, did not involve impassable terrain where no defensive preparations were thought to be needed.
How about the Battle of Quebec? Montcalm considered the cliffs unscalable, and was therefore quite shocked when Wolfe’s grenadiers did just that.
This isn’t what the OP asked for. We want examples where terrain was considered impenetrable, and was.
Oops. It wasn’t the OP at all, but Pochacco who brought up this point. Sorry.
(sneaks away, trying to look inconspicuous)
I think every type of terrain you can get people onto or into has had a war fought on it, even if it’s just between a few diseased, starved, frozen survivors of the respective armies. In really bad terrain though, the defences tend to be set up on the edge to force the enemy to attack across it, in the expectation that an adequate defence can be mounted using three grandfathers armed with a broom handle and a bucket of dried potates between them. Sometimes the defenders get the calculation wrong (e.g. Quebec), other times the attackers (Cartagena). Even if someone mounts a successful attack, it tends to be very expensive.
I think you have to place Kokoda Trail quite high up in the ‘ooh, impressive, but you really didn’t wanna do that’ list, plus the jungle fighting in Burma, with special mentions for Kohima/Imphal and the Chindit expeditions. Terrain that was initially regarded as impossible, was overcome after a fashion, but absolutely shredded forces traversing it. Sending a regiment through the jungle so four or five companies would arrive in a fit state to engage the enemy proved a bit of an expensive proposition. Casualty rates would hit 30% or more, with many of the survivors fit for nothing but medevac. Also check out the Italian front of WW1 as some great terrain to avoid in winter.
Yeah I’d agree that for every instance of “unpassable” terrain proving passable (Hannible crossing the alps, Invasion of France through Ardennes), there are plenty of instances of “unpassable” terrain proving just that (Ottoman invasion of Russia in WW1 being a prime example)…
Though of course that doesn’t really answer the OP. What would be more relevant would be defeats due to the defenders spreading themselves too thinly where they could have concentrated themselves in the regions that did have passable approaches. I can’t think of any off the top of my head.
Unless you’re talking about crossing the Atlantic Ocean by foot, I don’t think there are any cases of a terrain being literally impassable - individual travelers have crossed the Himalayas, the Sahara, the Amazon, and the polar caps. When people say a terrain is impassable they mean a group can’t cross it as an effective military unit and that’s what happened in the German forest. Obviously if the forest had been literally impassable, Varius wouldn’t have lost his legions - they’d have gotten to the edge of the woods and then had to turn back and return to Rome.
Of all the frontiers between Germany and France, the Ardennes mountains were considered impassible, by the French high command. Oddly enough, that is where the German army swept in…and broke through the French lines.
I’ve always been puzzled by this, because it is not such difficult terrain. True, it is a mountainous region, the the mountains are low (most < 1000 meters), and is broken up by ravines and steep river valleys. And it is not very large as well. The other thing: the French could have used the terrain to their advantage-occupying the high grounds above the roads would have stopped ant German tank advance cold-you just needed determined troops equipped with modern anti-tank guns (which the French had plenty of). But no-the rench sent their army into Belgium, where it was surrounded and outflanked.
Truly one of the big mysteries of WWII.
Remember that the French high command decided this about 1925, shortly after WWI.
And they were probably right at the time–the Ardennes was impassible – for the tanks of the 1920’s.
But tank design & construction improved quite a bit over the next 15 years.
Without any matching improvement in the thinking of the French high command, apparently.
But that is not a big mystery of war! The tendency of military high commands (from any country) to refuse to change their preconceived plans to take into account new technology is legendary. Part of the reasoning behind the comment that ‘generals are ready to fight the last war over again, but do it right this time’.