If it's a nice summer's day at lowest avalanche warning level in mid altitude you can still be hit by an avalanche and die. That being said in reference to the answer by @BenCrowell and the comments, it's just a matter of chances which are relatively low to get AMS in regions up to 3000 m or even slightly above.
With higher altitude the partial pressure of oxygen in the air decreases. While in a long-distance flight in about 10 km height the conditions are similar to air in 2000-2500 m altitude. With the decreasing partial pressure of oxygen in breathing air the partial pressure of the dissolved oxygen in your blood decreases. The oxygen saturation drops to about 90% for this height. Related to your condition which is affected by your predisposition or actual stress factors which often occur on alpine tours, this decreased oxygen level will be balanced by an increased breathing rate and you will take deeper breaths. For some people this isn't enough and they already notice the effect of the decreased oxygen saturation on their body very intensively.
It is proven that mountain tribes like the Sherpas have bigger red blood cells (hemoglobin) in consequence to their habitat. This is very admirable in my opinion and it shows the diversity of human being and the beauty of evolution. And like you know it's a major difference if you go from 1000 to 3000 m or if you start by sea level. The key is acclimatization.
I occasionally read about altitude sickness in Hochtour which is a quite new German book about alpine tours. I will share some general information I was getting from the chapter relating to AMS (acute mountain sickness):
- occurs mostly in altitudes up to 6000 m
- characterised by continual dull tapping headache
- other symptoms are tiredness, weakness, nausea, anorexia, sleep disturbances, apathy, distinctly increased pulse at rest
- up to 2500 m: under normal circumstances no AMS because the adaption of the body
- 2500 - 5300 m: adaption of body isn't enough; can achieve the complete performance by proper acclimatization
- above 5300 m: complete acclimatization isn't achievable anymore; limited to temporary stay
How to avoid AMS?
- above 3000 m: increase overnight stay by maximal 500 m per day or 1500 m per week
- stay 2 nights at a height if you can't avoid big differences
- observe your pulse and respiration: only use approx. 50% of maximal performance while acclimatization
- if pulse at rest is increased by 20% in relation to normal (valley) pulse you are still acclimatizing and need caution (do lots of breaks or even a rest day)
- can use hyperventilation to increase partial pressure of oxygen in the lung
- don't ignore first warn signs; speak with your team (not only important for AMS related questions)!
Don't panic if you are getting headaches in higher altitudes. They are not unusual above e.g. 4000 m. If they are occurring together with other symptoms you should start to worry a bit more. Be cautious, listen to your body and ask your mates because sometimes you aren't able to judge by yourself anymore.