Do particles gain mass only at energy levels found during the big bang? I am trying to make sure my understanding is correct. 
At energies and temperatures found during the big bang (or at CERN recently), the Higgs mechanism comes into effect. When it does, there is a break in electroweak symmetry (because the field is unstable?) and a condensate is formed. This condensate is what allows bosons to gain mass.
So at the moment of the big bang, all the bosons (and fermions I suppose) were massless (due to symmetry), and then at some point after the big bang when the temperature of the universe was just so, the Higgs mechanism took hold and gave the particles mass.
Is this correct?
 A: Not quite. The Higgs mechanism actually applies at low energies. Don't think of it as an event that happens once and bestows mass upon all particles for the rest of time; instead, the Higgs mechanism is a continuous effect that explains how particles are able to have mass at low energies.
For a full(er) explanation, I'll point you to another answer of mine, but the gist of it is that the broken symmetry that produces the Higgs mechanism is a symmetry of the Higgs field. If the field is $\phi$, it has an associated potential energy density $-a|\phi|^2 + b|\phi|^4$, which has a maximum at $\phi = 0$ and a continuum of minima along a ring at $|\phi| = \sqrt{\frac{a}{2b}}$. Early in the universe, the field would have been excited to much higher energies than either the central maximum or the ring minima, but as the universe cools and the energy density drops, the field "settles" into some minimum along the ring.
A: The Higgs mechanism is a mathematical "sleight of hand".
The physical idea is that "empty" space is an electroweak superconductor after the Higgs field acquires a vacuum expectation value which it must have when in a ground state.
So, regardless of the energies and temperatures, if the Higgs field has a non-zero VEV, gauge and matter fields appear massive "inside" the electroweak superconductor.    This is analogous to how photons appear massive inside an electromagnetic superconductor.
Of course, if the average energy are very high, the gauge and matter fields are effectively massless; the kinetic energies dwarf the mass-energies.
