Turns out the biological lens of your eye blocks UV light, but if you get an artificial lens, your retinas can register UV light.
There's some natural variation....
This has been understood for some time. As others have mentioned, various military orgs have used teams with varied color vision as a way of "seeing through" camouflage. Biologists have suggested that the variety in human color vision is adaptive, giving our hunting ancestors' teams an improved chance of spotting spotting prey against various backgrounds, and the addition of dogs (with their very different color vision from ours) improved this teamwork. This is all hypothetical, though, since (as far as I know) it hasn't actually been tested scientifically.
Back in high school (in the 60s), I had a science teacher who did a good illustration of it all. He made the usual demo of a spectrum using a prism, on a sheet of white paper. Then he had students come up and mark the visible ends of the spectrum, covering up each student's marks with another sheet of paper before the next student made their marks. The result was two columns of dots that didn't line up at all; their variants was around 10% of the width of the spectrum. I'd made marks that I could identify, and saw that my UV mark was right at the average point, while my IR mark was one of the farthest out. This explained some things I'd already noticed about the ways that different people saw colors.
This has been known to the photography industry since color film was first produced. Different varieties of film (and now CCDs) have different sensitivities, and different photographers have different preferences for brands of film based on this.
One of my funny personal anecdotes on the topic was once (in Jr High, as I recall), I asked some visitors why the front-left panel of their car was a different color than the rest of the car. They gave me a funny look, then said the car was all black, which everyone else present agreed with. I objected that only that one panel was black; the rest of the car was a deep red. This got me more funny looks, and the fellow who owned the car said that the car had been in a minor accident that damaged the front-left panel, so it was replaced. After that, my family thought I had something called "black-red color blindness" (which is odd, because I was actually the only one without that defect ;-). I was taken to an optometrist, who verified the "condition", but assured my parents that it wasn't a significant problem, and didn't need treating. Actually, there was a simple treatment: glasses that block near-IR light, and I've accidentally got several sunglasses that do just that, making for oddly muted reds.
As I got more into photography, I eventually noticed that my eyes have slightly different color vision, with things looking slightly bluer in the left eye and slightly redder in the right eye. This seems to be extremely common, actually, though most people don't notice it until it's mentioned and they start trying to spot it in different lighting condition. (Hint: It's often easier to spot in lower-light conditions, and difficult in full sunlight.)