What are the conditions for a planetary moon to have a satellite of its own? How far do the Solar System's bodies fall from the necessary threshold?
This is due to the mechanics of capture.
For one object (moon) to be captured by a another (planet), some energy has to be removed from the system. If the incoming moon has an existing satellite then it would be ejected, carrying a lot of kinetic energy.
If a small body were to be captured by the planet/moon combination, it would usually be captured by the larger object. Again some energy has to removed from the incoming object by being transferred to one the objects already present. One way that could occur is for the existing moon to be accelerated into a higher orbit.
In summary, it could happen, but with a very low probability.
For much more information, see this answer to the question How do moons get captured?.
In addition to the capture mechanisms mentioned by Andy, you also need to take into account the stability of any orbit due to perturbations from tidal effects. E.g. in case of our Moon, there are no stable orbits possible. Every satellite put in orbit around the Moon had had to implement course corrections to prevent it from prematurely crashing into the Moon.
This effect explains why you don't have a lot of dusts and rocks in orbit around the Moon due to the large number of impacts. What happens is that after an impact you do get a lot ejecta, a small fraction will then collide with each other so that they get in an orbit around the Moon. But on the long term the tidal effects of the Earth and the Sun causes perturbations until the orbit crosses the lunar surface.