Why are trichromat cone cells unable to sense ultraviolet and infrared radiation?
The range of color perception in humans is primarily limited by the sensitivity of cone cells (specifically, the opsin proteins in cone cells) to various frequencies of light.  This sensitivity drops off at 700 nm on one side and 380 nm on the other, which is where infrared and ultraviolet begin, respectively.  

Even with hypothetical infrared or ultraviolet cones, we wouldn't necessarily be able to perceive color much further beyond our current range anyway.  The various parts of the eye (cornea, lens, and the liquid inside) block both ultraviolet (starting around 400 nm) and infrared (starting maybe at 1400 nm) beyond certain wavelenghts. 

http://en.wikipedia.org/wiki/Color_vision

Tetrachromaticity, at least in humans, does NOT confer an extended spectrum of color perception.  Tetrachromaticity results from a mutation on one of the light sensing proteins, changing its peak sensitivity by a few nanometers. So a tetrachromat can't see beyond the normal color spectrum, but they can probably distinguish shades of green and blue more precisely than trichromats. This has been an area of active scientific research recently; I think most scientists didn't think they existed.  Now studies suggest they're 2-50% of the population.  Humans with 5+ cone types are also hypothesized, and with the ubiquity of genome sequencing I imagine we'll be learning a lot more about that soon.

Other species have more diverse color perception, with ultraviolet being especially common in insects and birds. The mantis shrimp has an astounding 16 different kinds of photo receptors, plus a few specialized eyes for detecting polarized light. For a loving account of mantis shrimp, I recommend this comic. http://theoatmeal.com/comics/mantis_shrimp