Yes, it is, but that won't help much.
In the case of diamond, it was formed by subjecting some low-density form of carbon (organic or inorganic) to high temperatures and pressures, transforming it into a higher density form. Typically, it starts out as some form of coal or organic compound with a density near 1g/cc and forms diamond with a density near 3 g/cc. What's important is that the raw material is an amorphous material in which the atoms are very inefficiently packed, and the final product is a crystal in which the carbon atoms are more closely packed. Still, the crystal structure of diamond is such that the atoms are not packed in the what is known as a "closed pack structure". And with Carbon atoms, due to their electronic configuration, a closed pack crystal structure is not stable. If it were, you'd be able to get a denser allotrope than diamond, but even then, it would only have a density of about 5 g/cc (compared to Os, which is about 22 g/cc). The "problem" with diamond is that it is made up of very light carbon atoms.
To get higher densities, you pick more massive atoms with the right number of electrons that allow them to form crystals with closed pack structures. Os and Ir are good examples where both these effects are utilized. But having already picked an element that exists in a closed pack structure, you can not further increase its density by compressing it into a more efficient structure: there is no more efficient packing than a closed pack crystal.
The best you can do, is fill the tiny gaps (called interstices or interstitial voids) between the big Os or Ir atoms with some smaller atom that will fit in it. If you try to stick in atoms that are a lot bigger than the size of the voids, you first find that they increase the volume of the crystal (thereby reducing density), and then, with bigger atoms, they simply cause the crystal to rearrange into some other, less efficient structure.
Further reading:
http://departments.kings.edu/chemlab/animation/packgeo.html
http://departments.kings.edu/chemlab/animation/clospack.html
PS: This discussion is entirely about materials that are generally stable under terrestrial conditions, and therefore, does not get into exotic scenarios like degenerate neutron matter.