You don't see it directly, but you see its interaction with eye movement. See "phantom array" effect. I can see this up to at least 5kHz under the right conditions.
When there's multiple flickering LEDs in the same room, they create interference. So even if we don't notice a single one flickering, when all of them go from black to full at the same time, and this happens 20x a second, we can notice that.
While there are certainly conditions high frequencies (tens of kilohertz) can be detected by a human, this explanation isn't it.
In general, light is linear and time invariant. That means in a flickering light system, a flicker of blue light then of red light, or red and blue at the same time, are indistinguishable if the 'sampling' system doesn't have sufficient bandwidth to resolve the flicker at all.
There do exist nonlinear optical components[1], but none that exhibit nonlinearities at the light intensities you'll find at home!
Additionally, even if the above were not true, flickering LED sources tend to flicker at a multiple of the AC supply, so it's very unlikely you'd see one flickering at 1000 Hz and another at 1001 Hz for example. For battery powered devices, it could easily happen though.
Plenty of cheap LED light systems flicker at 50 or 60Hz, see elsewhere in this thread. Also, the flicker can be indirectly visible, for example by turning motion blur into a sequence of discrete strobe events. My favorite experiment is to shake the end of a spoon (because it is curved and reflective) back and forth in an area predominantly lit by strobing LEDs. You will see a number of spoons, instead of a blurry spoon like you would under incandescents.
kHz? Really? That seems very high to me.