This paper does not deal with cockroaches running, as they sometimes seem to do, at a speed close to that of light. Rather its subject is a cockroach nest, i.e., a normally innocuous crack in the wall, but from which cockroaches start pouring out if some food is put near it. The physical problem concerns first the possible values of the energy of two relativistic free particles. An elementary classical calculation, in the center of mass frame of reference, shows that energy levels E appear at E ≥ 2mc2 and E ≤ −2mc2but also at E = 0. The latter is our crack in the wall. Turning to quantum mechanics for two Dirac free particles we obtain explicitly the infinite number of states present at E = 0. We consider then a Poincaré-invariant two-body problem, along the lines suggested by Barut, and see how we can pass from two free particles, to the ones in which different types of interaction are present, i.e., put food near our cockroach nest in the wall. In particular when the two particles have a Dirac oscillator type of interaction, we can see exactly that in some cases the cockroach nest remains inert, in others, the cockroaches pour out, i.e., levels start pouring out from the one with E = 0. A variational procedure allows us to carry out numerical calculations for a particle–antiparticle system with Dirac oscillator interactions as well as what we call "positronium." In both cases all the cockroaches come out of the crack, i.e., all the degenerate states abandon the original E = 0. In the conclusion we indicate that this type of phenomenon also appears in the relativistic Hamiltonians employed by atomic physicists, and that the awareness of its existence may be of use in these types of calculations.