We already can do this in materials--- it is called "superconductivity". The phenomenon for photons was understood before the phenomenon in the weak interactions, and the description of superconductivity by Landau, and the Bardeen Cooper Schriefer model for fermionic paired condensates, was the inspiration for Nambu's fermion vacuum condensate idea, and Brout and Englert's later point-particle superconducting Higgs mechanism.
Photons do not go slower in a superconductor, they do not go at all. Superconductors don't have photon excitations at all, and if you have an electric and magnetic field in the superconductor trying to propagate, the fields decay away exponentially.
It is certain that we won't be able to do this in vacuum, because we know all the fields around us are stable. In order to make an instability in the field, we have to alter the fundamental constants in such a way that a charged field makes a superconducting condensate. In order to alter the constants, we would need a certain energy density per unit volume which is going to be practically infinite for the purposes of engineering.
But the condensed matter analog, the superconductor, is a perfect analog, and we understand the dynamics of what would happen in this situation simply by examining what happens in a superconductor, and extrapolating to the situation where the material doesn't break Einstein's relativity invariance with respect to constant motion.
If basic symmetry and homogeneity assumptions about the Universe hold, then yes, all massless real particles (see Anna V's answer for virtual particles must travel at a universal constant $c$, the speed of a massless particle, in all frames of reference.
Given these basic symmetry and homogeneity assumptions, one can derive the possible co-ordinate transformations for the relativity of inertial frames: see the section "From Group Postulates" on the Wikipedia Page "Lorentz Transformation". (Also see my summary here). Galilean relativity is consistent with these assumptions, but not uniquely so: the other possibility is that there is some speed $c$ characterizing relativity such that $c$ is the same when measured from all frames of reference. Time dilation, Lorentz-Fitzgerald contraction and the impossibility of accelerating a massive particle to $c$ are all simple consequences of these other possible relativities.
So now it becomes an experimental question as to which relativity holds: Galilean or Lorentz transformation? And the experiment is answered by testing how speeds transform between inertial frames. Otherwise put, the experimental question is are there any speeds that are the same for all inertial observers?. The question is not about measuring the values of any speed, but rather, how they transform. Now of course we know the answer: the Michelson Morley experiment found such a speed, the speed of light. So there are two conclusions here: (1) Relativity of inertial frames is Lorentzian, not Galilean (which can be thought of as a Lorentz transformation with infinite $c$) and (2) light is a massless particle, because light is observed to go at this speed that transforms in this special way.
Notice that at the outset of this argument we mention nothing about particles or any particular physical phenomenon (even though special relativity's historical roots were in light). It follows that, if $c$ is experimentally observed to be finite (i.e. Galilean relativity does not hold), then the specially invariant speed is unique: it can only be reached by massless particles and there can't be more than one such $c$ - the Lorentz laws are what they are and are the only ones consistent with our initial symmetry and homogeneity assumptions. So if we observed two different speeds transforming like $c$, this would falsify our basic symmetry and homogeneity assumptions about the World. No experiment gives us grounds for doing that.
This is why all massless particles have the same speed $c$.
Update: Experimental Results
As is now common knowledge, the gravitational wave event GW170817 and gamma ray burst GRB170817A give strong experimental evidence of the equality of the speeds of light and gravitation. As discussed in:
Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A
the 1.7 second time delay between the gravitational wave arrival and the gamma ray burst, together with conservative assumptions about other sources of delay, yields an experimental bound on the fractional difference between the speed of light and of gravitation:
$$\frac{v_g-v_{em}}{c} \leq 3\times 10^{-15}$$
an impressive experimental bound indeed. Within the next 10 years, we probably shall see several such events, and thus this experimental bound will tighten further (unless something really theoretically unforeseen happens!).
Mass From Confined Massless Particles
Incidentally, if we confine massless particles, e.g. put light into a perfectly reflecting box, the box's inertia increases by $E/c^2$, where $E$ is the energy content. This is the mechanism for most of your body's mass: massless gluons are confined and are accelerating backwards and forwards all the time, so they have inertia just as the confined light in a box did. Likewise, an electron can be thought of as comprising two massless particles, tethered together by a coupling term that is the mass of the electron. The Dirac and Maxwell equations can be written in the same form: the left and right hand circularly polarized components of light are uncoupled and therefore travel at $c$, but the massless left and right hand circular components of the electron are tethered together. This begets the phenomenon of the Zitterbewegung - whereby an electron can be construed as observable at any instant in time as traveling at $c$, but it swiftly oscillates back and forth between left and right hand states and is thus confined in one place. Therefore it takes on mass, just as the "tethered" light in the box does.
Best Answer
The mechanism for "giving mass" to elementary bosons and fermions is different.
With bosons, it is related to the gauge symmetry ($SU(3)_c \times SU(2)_L \times U(1)_Y$) which is partially broken (and become $SU(3)_c \times U(1)_{em})$. The unbroken part imposes its associated bosons (gluons and photon) to be massless to respect this symmetry.
With fermions, there is no such constraint since their mass does not come from a gauge symmetry (with our current knowledge, fermions masses are put by hand via add hoc yukawa couplings). Therefore, the mass of the fermions is not predicted (contrary to the masses of bosons). So, asking "why do we see no fundamental massless fermions?", is equivalent as asking "why do we see fundamental fermions with their actual mass?". Answer: we don't know!