How long does one have to live at altitude to train hard and race at altitude?

Depends somewhat on the elevation, but 3 weeks to feel okay, 6 weeks to race decent, and 3-6 months for full adaptation and maybe even more.

When I moved from about 1000 feet to 6000 feet I felt like it took me 6 months to feel okay. Still was not going be running as fast as at sea level but I stopped feeling like crap on the harder runs.

I did a 3 month stint at moderate altitude, 6500 feet. I felt it didn't help at all and actually impeded my training as I was up there in summer and when I came down humidity and heat killed me.