Getting above 10,000 feet produces altitude sickness (headaches, nausea, general malaise, and at the extreme, HAPE and HACE). There is an article in today's Wall Street Journal about research that is being done on AMS (Acute Mountain Sickness). You may have to be a paid subscriber to the on-line version of the Journal to see the full article, though it may be in the "free" section. It notes that a certain number of people have 6 genetic markers that indicate that these people are more readily adaptable to altitude. About 10 years ago, I participated in a study, and it turns out that I am among the lucky folks. This doesn't mean that those of us with the right genes can be dropped on the summit of Everest, or even a lower peak like Whitney or Rainier without suffering altitude effects, just that we adapt more rapidly. The basic rules for adaptation are (1) climb high, sleep low, (2) when above 10,000 ft, move your sleeping altitude up an average of 1000 ft a day (you can climb 2000 ft during the day, then come back down 1000 ft for your campsite, for example), and (3) if you start getting the headaches, nausea, etc, and especially if you have signs of HAPE or HACE, descend, descend, descend!
There are medications that will aid adaptation for most people. According to the article, acetazolamide (Diamox) and sildenafil (Viagra) work, but can interact badly with other medications and have side effects, like all drugs (Diamox's most infamous side effects are "the tinglies" where your fingers feel like they have fallen asleep, and carbonated beverages taking on a bad taste, most important for climbers being that beer tastes terrible). There are also some drugs that will counteract HACE and HAPE, if they have not progressed too far.
Although not mentioned in the article, a certain percentage of the population is genetically unable to adapt to altitudes over 9000 or 10,000 ft. I know several people in this category, one of whom is my primary care physician (as the insurance company term goes), much to his disappointment, since he and his wife are avid outdoor types, who we see from time to time backcountry skiing below the 8000 foot level.