Assuming that I had wood and a bunch of matches, could I get a fire going?
From what I have read it is possible but it would be very difficult. Temperature, low oxygen and so on. You’d need almost perfect conditions (completely dried wood, no wind and so on) and it would be a weak fire at best.
In short, yes. If you can be there as a human, and be awake/alive breathing the amount of oxygen that’s present, there’s enough oxygen to support combustion.
At sea level, things for people start getting wonky at 19.5% or less oxygen in air, and Very Bad at less than 10 or 11%. Fires can keep doing their thing down to 5% or less. At altitude we get into the partial pressure thing, but it’s still the same amount of oxygen relative between the fire and the human.
Jet planes go considerably higher than the summit of Mount Everest, and they burn fuel, so combustion must be possible - unlike rockets, jet engines rely on atmospheric air for combustion and don’t bring their own oxygen supply with them. Yes, I know, they compress air before injecting it into the chamber, but AIUI that is so they can burn more fuel at the same time - chemically, combustion would also be possible with uncompressed air.
I suppose by now it’s OK to share this story, about a Chinese paraglider who accidentally got sucked up to Everest-esque altitudes without oxygen supply and survived: Chinese paraglider survives accidental flight in the clouds without oxygen, reports state media | CNN
Sure. But jets use compressors to compress the thin air coming in to much higher pressures where combustion is easy.
If you have ever driven an automobile to high elevations you can easily discern them not working as well since they don’t have compressors (unless they have a turbocharger or something but not sure those are meant to compensate for high elevations).
Back when I took volunteer firefighter training, I thought I was told that fire and humans both needed around 16% oxygen. Admittedly, that was 30 years ago so I could be misremembering.
Your academy taught you mostly right. Flaming combustion starts to reduce around 16%, and peters out around 10%. Below that it’s generally smoldering combustion. Still burning, though.
That’s how and why backdrafts can be a thing - crappy combustion still takes place in a reduced oxygen environment, and the byproducts from that crappy combustion will support flaming combustion if we dial the oxygen back up.
As long as stuff is hot enough to pyrolize, it’s still generating readily combustible fuel.
Jets operating at high altitude aren’t a good demonstration of burning fuel at low ambient pressure.
Jets need air compression to burn fuel efficiently enough to produce thrust. In the case of a turbo jet combustion has to be sufficient to drive a compressor. Ramjets need velocity produced from thrust to compress air in the combustion chamber and pulse jets rely on their pulsing exhaust to compress air in the combustion chamber between combustions.
Surely, it’d depend on what fuel you’re trying to burn?
Wood and matches according to the OP. Not that hard with a high enough ratio of matches to wood.
However, aircraft getting shot down at altitude do demonstrate fuel burning at ambient pressure, and there are lots of examples from WW2 onwards. An incendiary round or 2 to a fuel tank is going to ruin your day.
But that does require an intense source of heat to get the fuel burning.
Thought I read somewhere that commercial airliners somehow keep the cabin oxygen at 10%(?) and can dial that back at will to render, say, a rambunctious rugby team unconscious.
Specifically, fuel-burning automobiles.
Pressurized airplanes generally maintain either the equivalent of 5,000-8,000 MSL pressure due to compressing the air flowing into the cabin. I’m not aware of any that can control the ratio of gasses, it’s just what the ambient has but at a higher pressure.
Very much doubt any pilot is going to “dial down” the pressure to render passengers, even rambunctious ones. The liability would be daunting if something went seriously wrong, and it would probably set off all sorts of alarms they’d need to explain.
Once they get that Everest elevator installed, we can test this fire question.
Appreciate the info.
Cripes, I probably read that in a Stephen King story and assumed he researched it better. 
IIRC it is the equivalent of 8,000 feet for most passenger planes. The B-787, because it is built from stronger materials, will pressurize to the equivalent of 6,000 feet (so, a higher internal pressure). Much more comfortable for the passengers.
Am I the only one who has on more than one occasion failed to start a fire with just matches and wood at sea level? I guess starting a fire on top of Everest is technically possible; but how likely is success, especially if your only kindling is a bunch(?) of matches?
Well, there is always a non-zero chance the fire will fail to ignite. Normally I’m a reliable 1-match campfire gal but sure, I’ve had failures.
Using just matches as kindling is far from optimal, even at sea level. As noted up thread you’d need to bring something burnable with you as rock and snow are not notably flammable.