From Nathan's and my experience, I believe kids are capable of a lot more than we often give them credit for. The key is to be sensitive to their condition and flexible enough in your plans to react quickly if they do have problems.
We live at about 500' near St. Louis, so we start from scratch on any of our mountaineering trips. Nathan went to Crater Lake, Oregon, when he was about 2-1/2 (only between 7,000' and 8,000', granted) and hiked the trail from the rim down to the lake and back up under his own power. That's about a mile each way and 700 or 800 vertical feet.
Then when he was 5-1/2, we went up to Mt. Ritter, camping above Ediza Lake at about 9,500' and wandering around up to at least 10k. At 6-1/2, we went out to CO and rode the cog up to Pikes Peak, about 14,100'. We had already been out to Aspen and over Independence Pass at that point, so he was well acclimated.
When we did Whitney in '01, he was 12. We spent two nights at Mammoth Lakes (8,000') and did a day hike up Glass Mountain to about 11k in between, then headed down to Whitney and hiked up to Trail Camp (12k) before summiting the next day.
Acclimation is certainly important for all of us, kids probably more so. As long as the child is old enough to tell you accurately where it hurts, and you are watching for potential AMS symptoms, by all means, take them along...they'll slow you down a bit but the enjoyment in their sense of accomplishment is more than worth it and you'll set them up with a love of the mountains that many of us didn't discover until we were adults.
Trip reports for many of our meanderings are on my
WWW site.