Why are there only three spatial dimensions and only one time dimension?
Perhaps it is only our perception, and through a fourth dimension they are nearer to us.
In pretty much all of physics, if the number of dimensions is different from 3+1 (spatial and time), things completely fall apart, and all our physical models break. This is true of everything from the geometry that the Greeks and Egyptians were already doing, to general relativity (which includes gravity as a subset), and quantum field theory (which has means the standard model of particles, which includes quantum mechanics as a subset).
(Side remark: String theory is often expressed as a model that operates in 10 or 11 dimensions, but those extra dimensions are "rolled up" by the Kaluza-Klein theory, and are inaccessible at energies below a threshold, which is far away from anything we've ever experienced or seen or have any way to measure (even indirectly). And note that our experience is now reaching very high energies, for example with the 200 PeV neutrino detected in the mediterranean recently. So these extra dimensions are guaranteed to be non-interacting with the physics that determines our daily life and our cosmology, perhaps except a fraction of second after the big bang.)
Is there a way to directly measure that we have 3.0 +- 0.1 spatial dimensions? Not easily, because the mathematical model is so closely built around using that. We can very accurately measure the number of quark species (from the hadronic R in e+e- colliders), and the number of neutrino flavors (from the decay width of the Z0 boson). The best experimental proof of the number of spatial dimensions comes somewhat indirectly, from the equipartition theorem of statistical mechanics (itself a close relative of the virial theorem), which correctly predicts the ideal gas law (with the famous factor of 3/2 k T, where the 3 is the number of dimensions, and k is Boltzman's constant), and the heat capacity of solids (I forgot the exact name for that law, might be Dulang-Petit). The argument behind that is that a molecule in an ideal gas has a certain number of "degrees of freedom" (ways it can move), and the amount of energy in each of the degrees of freedom will be the same, so by measuring the molecules kinetic energy (which is an aspect of the gases temperature) you get the number of dimensions it can move. So indeed, any experiment in thermodynamics that involves a gas pretty much measures that the number of spatial dimensions is 3.
Obviously, we can create theories (better called "science fiction") that have more dimensions, and we can hypothesize that space aliens have ways to access those extra dimensions. But such ideas are wholly outside the realm of experimental physics, and are more a branch of literature and social criticism.
