We all know when people run out of O2 up there, trouble follows. Is there a breathing technique to get more mileage out of bottled oxygen?
Let's stick to the basic scenario of the need of bottled oxygen: high altitude.
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It depends a bit on the system used.
If you are being supplied with air or oxygen via demand valve then your consumption will depend on your rate or breathing rather than the amount of oxygen biologically needed by your body. So clearly in this situation being able to control your breathing rate will help a lot. Humans, like most animals have a natural response to danger to breathe deeper and faster, this is based on the 'assumption' that there is plenty of air available and you want to suck as much through your lungs as possible to prepare for 'fight or flight'.
This is fine when you are on the surface of the earth where humans evolved but is not the best response when you have a finite supply of oxygen available. Clearly there is no simple answer to this but it certainly makes sense to keep your level of activity stable and as efficient as possible.
It is a bit different though if you are breathing from a closed system eg a space station or re breather. In this case any exhaled oxygen is still available to be used as long as the carbon dioxide concentration can be kept within safe limits. In this case your oxygen usage is directly linked to your metabolism so anything you can do to conserve effort will help.
The third scenario is that oxygen is supplied to a semi-open mask at a constant rate in this case there is little or nothing you can do to conserve it so you are best off breathing as much of it as possible to store it in your blood (to the limited extent that this is possible) before it runs out. Clearly here understanding of your specific equipment and the way your own body responds is vital to getting the most out of it.
It is also worth adding that the human breathing reflex is based on Carbon dioxide concentration in the blood rather than oxygen and this can lead to counter intuitive problems in some situations. For example if you are exposed to high concentrations of inert gas like argon you can suffocate because you 'forget' to breath.