It's a good question. The answer is that the bound on the density is given by the requirement that the interactions between the bosons have to remain weak for the Bose-Einstein condensate to exist. In practice, the helium-4 atoms have to be further away from each other than their radius.
Why it is so? Well, if you're talking about the bosons occupying the "same state", it really means that you are constructing a multi-particle state in the multi-particle theory. If you want the energy of this state to be simply given by the sum of the energies of the individual bosons - i.e. $N$ times the energy of the one-particle state - you must guarantee that you have the right Hamiltonian which is essentially the Hamiltonian for the bosons in an external potential, without any significant interaction term in between the bosons.
A sufficiently strongly interacting Hamiltonian for the bosons couldn't be solved that easily.
If you try to push the composite bosons really close to each other, i.e. by lowering the temperature extremely close to the absolute zero, the interactions between them will start to matter which will prevent you from approximating the Hamiltonian by a sum of many one-particle terms. Consequently, the right description is in terms of the component particles - which are often fermions.
It's believed by many condensed matter physicists that the ultimate state of any bound matter very near the absolute zero is a superconductor or a Fermi liquid - and I don't know. Consider helium-3 as an example. I am actually not sure what one gets at superextremely low temperatures.