So, just for the heck of it, I thought I'd show this thread to a friend of mine who teaches engineering at the university level. Not electrical engineering, but he has enough grounding in it to deal with my level of questions.
In short, my math is correct, but only on an ideal level. I'm not taking into account things like the resistance of the wires, the kind of power that the proffie is supplying, thermal breakdown, and a couple of other factors. Ideally, I should be troubleshooting this with an oscilloscope, but as I don't have one or access to one (right now, at least) any answers I get to aren't fully going to resolve the issue, and any math I come up with is going to be very simplified. Although he does also suspect the LED might be dodgy, so, that's something.
He also ran up the datasheet for the crees, and thinks that there isn't much point running them much past 0.7A, because the brightness gained for the power consumed isn't going to change enough to be perceptible. Also, he doesn't get why everyone here suggests carbon film resistors as the only thing to use, and would like it explained better than "Well, 'Erv said to use them a decade and a bit ago, and it's kinda stuck." So, if anyone's got one, I'm all ears.