Most people that live at or near sea level will start to feel some of the effects of lower atmospheric pressure (high altitude) at about 5000 feet. This typically manifests as slightly elevated heart rate and respiratory rates as compared to your normal rates at lower elevations.
Altitude illness isn't really a concern until you are at or above 8000-feet, and even at that elevation, cases of Acute Mountain Sickness (AMS) are extremely rare.
At or above 10k, AMS becomes more common and could progress to HAPE or HACE.
There is some good experience-based feedback in the replies above. I'll try to offer a little more based on 20-ish years of walking around in the high mountains including eleven years leading trips on Denali at 20,310'. I've had the chance to see the effects of truly high altitude on thousands of people and there are definitely a few simple things you can do to help set yourself up for success with regard to adequate acclimatization.
As has been mentioned above, start slow. If you can spend a night or two an "intermediate" elevation on your way to the high country, this will allow your body to begin the compensatory process. AMS and the edemas (HAPE and HACE) are caused by a lack of oxygen in your blood and adequate perfusion. As you spend time at altitude, your body builds more red blood cells, thus increasing it's carrying capacity for oxygen leading to better perfusion and hopefully staving off illness. The altitude at which you sleep is more significant that the altitude that you hike to on any given day. The tried and true adage in the mountaineering world is "climb high sleep low."
If your target "high camp" is at 10K or close to it, spend a night at 5K and/or 7.5K and try to be somewhat active during these days. This is pretty conservative as plenty of people just blast right up and do fine, but why not take the time if you have it. I have definitely seen a number of folks go from at or near sea-level to sleeping at 10K and get sick. This is usually AMS that resolves in a day or two, but I'd rather spend those two days lower down acclimatizing rather than up high feeling like crap while trying to go hunting. I feel like I have seen this scenario play out a number of times, including having experienced it personally. A lowlander goes from at or near sea-level to 10K on day 1 and sleeps. The next day they hike around at slightly higher elevations (12-13K), get tired and dehydrated, and sleep at 10k or higher if they have moved up to a higher camp on day 2. On day 3, AMS comes knocking and rocks your world.
Altitude is the great equalizer as had been said above. Previous experience at altitude and/or physical fitness matter very little with regard to how well you will acclimatize. Every exposure is different and your body may or may not succed in compensating. That having been said, the more times you go through an acclimatization process, the more you will become familiar with how your body adapts which will allow you to keep better tabs on how you are doing and hopefully how to address problems before they arise.
Hydration and fatigue both play a role in how well you acclimatize. If you are tired and dehydrated it is a lot harder for your body to do what it needs to do, ramp up compensatory mechanisms. Stay active for your first few days, but don't run yourself into the ground too early and you will set the stage for a successful acclimatization process.
Recognize the signs and symptoms of AMS early. If you have a headache, high level of perceived fatigue, malaise, etc. go down ASAP. Dropping out for a day or two and coming back up is the best fix for AMS. Hanging at a given altitude hoping and trying to feel better and/or going higher is what can lead AMS to HAPE/HACE. The edemas are extremely rare for people sleeping below 14K, but it can happen. Severe AMS is probably the most significant concern and a bad bout of that can definitely be a show stopper.
Regarding prescription drugs, Diamox (acetazolamide) is a great drug, but it does have some side effects that are a serious concerns for people engaged in any sort of high output activity like climbing, mountain hunting etc. In a greatly simplified nutshell, Diamox makes you breathe more. As such, it aids the acclimatization process by staving off AMS symptoms while your body compensates. as has been mentioned, it is a diuretic which is a big concern re: hydration and rest (getting up to pee all night), both of which are important for an acclimatizing body. The dose that will likely be recommended by the doc is one 500 tablet twice a day, which is a huge dose. Lots of studies in the mountaineering world have demonstrated that half or even a quarter of a tablet one time a day provides essentially the same benefit as the full course with greatly reduced side effects.
My personal philosophy re: drugs is that I use them to treat symptoms as they occur rather than prophylactically. Having a bit of Diamox along in case you start feeling like crap on your first night at higher altitude is not a bad idea. It can be very helpful to ease symptoms and facilitate your immediate descent.
Dexamethasone and and Nifedipine are life saving interventions for HACE and HAPE respectively. Again these are used to treat symptoms and to facilitate immediate descent. These things should be considered in med kits for those planning to spend extended times at altitudes above 14K.
For trips that base at around 10-12K and top out at 14K, I typically don't take any altitude meds along other than vitamin I aka ibuprofen.
So in summary and in the context of what I would call a moderate elevation mountain hunt, I'd recommend the following:
1) Spend a day or two at intermediate elevations before you sleep at your higher elevation camp. If you plan to go up from there. Don't sleep more than 1000 feet higher per day, that is don't sleep at 12K until you've spent at least two nights at 10k, etc.
2) Remain moderately active for your few few days as you gain elevation and avoid any monster days aka excessive fatigue and dehydration for your first couple of days.
3) If you start to get sick, lose 2-3K ASAP, spend another day or two, and come back up when you feel better.
4) Be honest with yourself and your partner(s) about how you are all feeling. All of the illnesses described above start as relatively slow progressions which are easily solved early on. A daily self/team assessment and ongoing communication is key to maintaining a good group dynamic.
Good luck and enjoy the high country!
