Only if rapidly decompressed. Un-pressurised flight carries a very low risk that increases slightly as you ascend and usually will have had another trigger to cause it.
To put it in context, ascending from a depth of 3m to the surface while diving is the same pressure change as the surface to 18,000ft.
The reason that 18,000ft is given as the threshold for use of cannula is that your nostril area does not provide enough volume flow for inspiration of sufficient O2 for efficient gaseous exchange to occur. This is exacerbated by the fact that you will be talking etc and so end up starved off oxygen. Going onto a full face mask allows a great tidal flow of oxygen to be taken in through both the mouth and the nose. As altitude increases further the lungs no longer have enough dwell time for gas exchange to occur and forced oxygen breathing systems then kick in.
It is nothing to do with experience or knowing what you are doing as has been suggested. Neither will you begin to feel 'unwell'. Hypoxia is an insidious killer and generates feelings of Euphoria and well being not of illness. I have some interesting video of chamber trials that we have done over the years testing various bits of life support kit. The test subjects were adamant that they were in control at all times and were not feeling the effects of hyoxia. The video evidence to the contrary is quite enlightening.
I would caution those who are planning on the high altitude use of oxygen to seek training in the physics and physiology rather than accepting the word of internet posters on what works. I would also be very disinclined to trust a finger oxymeter for accurate O2 readings at those altitudes.
I've done an hour at 20k and a number of 4-5hr bits at 19k, with one passenger.
I thought the FAA mandated FFM above 18,000ft?