Bound State of Only Massless Particles? Follows a Time-Like Trajectory? Is there any way in which a bound state could consist only of massless particles? If yes, would this "atom" of massless particles travel on a light-like trajectory, or would the interaction energy cause it to travel on a time-like trajectory?
 A: John Rennie has answered the first part of the question. The second part was this:

If yes, would this "atom" of massless particles travel on a light-like
  trajectory, or would the interaction energy cause it to travel on a
  time-like trajectory?

The answer is that it would have a timelike world-line, and this is independent of any (probably uncertain) details of the system's dynamics or binding energy.
Mass is not additive. Mass is defined (in units with $c=1$) by $m^2=E^2-p^2$, where $E$ is the mass-energy and $p$ is the momentum. $(E,p)$ is the momentum four-vector, and the squared mass is its squared norm. For a massless particle, the momentum four-vector is lightlike. If four-vectors $p$ and $p'$ are both lightlike and future-directed, but not parallel, then $p+p'$ is timelike. Therefore a system of interacting, massless particles is guaranteed to have a nonzero mass.
A: Yes, or at least probably. Glueballs, if they exist, are bound states of two massless gluons. However glueballs (are calculated to) have a non-zero mass because their binding energy contributes to their mass, so they would travel on timelike trajectories.
I think the glueball is a special case because the strong force is confining. I'm not sure what would happen for a hypothetical massless system bound by the electromagnetic force.
A: The Standard Model, as validated by a large number of measurements, extrapolates all particles  to 0 mass at energies larger than symmetry breaking energies. I have never heard of bound states in the early cosmology models : with symmetry breaking, one gets  quark gluon plasma  and with further cooling the quarks bind into the protons etc as we know them.Before symmetry breaking the particles are free. One would have to propose a different theory for anything like this, which would manifest exactly as the SM at low energies; then also the interpretation of early cosmology would be different.
