DH has COPD and he had some problems at altitude. He did fine most of the time in NM, although we didn't go hiking in any mountains due to his low stamina. But he had a very hard time at the Grand Canyon South Rim, which is about 7,000 feet. In fact, he developed altitude sickness--thank God we had his oxygen with us! Still, he was pretty sick for a couple weeks after that.
About a month later we were in Yellowstone NP. DH by that time was fairly adjusted to altitude, although he was quite fatigued. He did fine most of the time, but when we went over the northeast loop, he became noticeably short of breath and lethargic. Again, oxygen to the rescue. He felt pretty bad, though.
Everyone is different. Perhaps your COPD isn't as bad as DHs. If you think you might have trouble, take your oxygen.