Bound states in QCD: Why only bound states of 2 or 3 quarks and not more?

There is no known reason that you can't have bound states like $qq\bar{q}\bar{q}$ or $qqqq\bar{q}$ or higher number excitations, but none have been observed to date.

You do have to make a color-neutral state, of course.

In the mid-2000 some folks thought that they had of pentaquark states (that the $qqqq\bar{q}$) for a while, but it was eventually concluded that they were wrong.

Added June 2013: Looks like we may have good evidence of four-quark bound states, though the detailed structure is not yet understood, and in the comments Peter Kravchuk points out that pentaquarks have come back while I wasn't paying attention (and the same state, too). Seem some egg may have moved from face to face.

As a quick explanation: all bound states are color-neutral. The intuitive reason is that the strong interaction is so strong that it would pull any color-charged particles together. (Because the strong force increases with distance, you can't get around this by spreading out the charged particles, as you can with the EM interaction.)

Since there are 3 colors, you can either achieve a color-neutral state by combining one quark of each color, which gives you a baryon, or a quark and an antiquark of the same color (e.g. blue and antiblue), which gives you a meson. Any combination of more quarks or antiquarks that works out to being color-neutral, such as the hypothetical pentaquark, can be broken down into some combination of baryons and mesons, which means that such a particle would probably naturally decay in that way, if it could even exist (which there is no evidence for).

In a sense every nucleus is a bound state of 3N quarks. After all, the nuclear force between nucleons (protons and neutrons) is a result of the leakage of the strong color force outside the "boundary" of the nucleon. So there are undoubtedly gluons and even quark exchanges between the nucleons of a nucleus.