Allision I had serious altitude problems when I fist hiked the JMT three years ago, but I fell in love with that area of California and decided I would do as much as possible for my trip back their the following year. My problems were headaches and waking up in the middle of the night convinced I was drowning.
I live a few hours from Toronto, Canada at an altitude of 1000 ft, in very flat farm land, and until going to the Sierra's I had only backpacked to a maximum of 8000ft. Training at altitude was just not an option for me.
As others have stated the headaches were easily solved thru altitude acclimatization and drinking extra water, I now make a point of drinking 3/4 Litre of water at mid day break, as well as my normal water consumption along with Acetaminophen.
Next I spoke to my doctor who ran blood tests, and ECG, and chest x-ray, all showed negative results. The MD thought it might be an allergy and gave me a puffer, but that also has shown to be negative.
I changed by arobic exercises, I had been doing Interval training on the cross trainer for 1 hour, I stepped this up by decreasing the time to 20-30 minutes but increased by maximium heart rate to 220 - age, I only reach this very high rate for a few seconds on the up interval and try to drop my heart rate by 30 beats in the down interval. I find this rather extreme exercise easier on the stationary bike.
A few more things I changed although they aren't related to training. I now take Ginko biloba twice a day at 60ml. Mountain climbers have had some success with Ginko and in Canada it does not require a prescription. I start the pill 2 weeks before the trip and continue to the trip is over.
I raised my head when sleeping by making my pillow much bigger. I now put the empty pack under the head of the Thermarest, and the pillow a Mount-Bell UL propped up by my clothing stuff sack velcroed to the mattress. The inflatable pillow is tied to the mattress by its front holes. I also take an extra hour for sleep to help with the altitude.
On my last trip on the JMT, I did an altitude acclimatization hike of a week. I did an easy hike through the Emigrant Wilderness area and kept the altitude below 10000 ft.
The night before doing the JMT this year I slept at Horseshoe meadows at an altitude of 10000 ft, and experienced very mild symptoms compared to my first night three years ago which was also at HM. I hope this long post helps.