Fairly straightforward question. If not, why not?
I suspect that if they do, it is not perceived due to the regions of highest dispersion being in one's region of lowest visual acuity.
Yes, it does. We don't see it because our brain automagically 'correct it' because it always see the same aberration from the childhood.
Our eye focuses on 'green' wavelength as it's its peak sensitivity, so red and especially violet lines are usually slightly out-of-focus.
Because refractive index is a function of wavelenght, every lens experiences chromatic aberration, and so does a human eye.
This could have severe effect on human vision. However, eye has tuned spectral bandwidth - it is well known eye is most sensible to visible light which has wavelenght of 550 nm. Relative luminosity is also a function of wavelenght (in which more than 70% of the luminous energy is confined to a defocus range of less than 0.25 D defocus on either side of focus (if eye is optimally focused at 550 nm)). The exact spectrum to which we are sensitive, depends on light levels. Rods saturate at moderate and high light levels, so in this region spectral sensitivity is governed by cones. At lower light levels cones are no longer as sensitive and rods dominate the response. All this results in different spectral sensitivity for various luminosity and in this way eye is able to filter out some effects of chromatic aberration.
Here is an interesting webpage on the aberrations of the eye: http://www.telescope-optics.net/eye_chromatism.htm
According to http://kaschke-medtec.de/SS10/Exercise5.pdf the eye has quite a large chromatic aberration of 2 dptr across the visible range.
I heard (the rumor?) that for this reason the color bands in the french flag don't have the same width.
I also read of the interesting (and probably true) theory that the colour bands around edges actually help you to focus while you reading text. I think that explains why I find reading quite hard with LED illumination of small (non-overlapping) spectral bandwidth.
One could think it would be useful to wear color corrected glasses but I don't think that would help in any way (the aberrations due to the glasses will be small compared to the aberrations in the eye).