What you're describing is a large part of what's known as the measurement problem, and it is quite certainly the main open problem in the study of the foundations of quantum mechanics. In the Copenhagen interpretation of QM, one has a quantum mechanical system on one hand and a classical measurement device, which implies drawing a line somewhere between the two. One typically hopes that it doesn't really matter where the line is drawn and so far it's been possible (always!) to draw it somewhere so results match experiments. However, there's no clear physical principle for where to draw it, and your question ("when is a system big enough to count as a classical measurement device?") does not yet have a systematic answer.
However, as our technology gets better, we've been able to make systems that push this line up and up (and in fact you can now place objects visible to the naked eye in quantum superposition states). For these ever-bigger states, increasing numbers of particles come to have ill-defined positions.
For large systems, however, it becomes very difficult to observe these superpositions because of decoherence. This is due to the fact that a Schrödinger's-cat state $$|\psi\rangle={1\over\sqrt{2}}(|\psi_1\rangle+|\psi_2\rangle)$$ is delicately sensitive to the relative phase of its two components. You typically detect these states (i.e. you discriminate them from a simple statistical mixture) by observing some kind of interference pattern, and if you change the $+$ for a $-$ then the interference pattern shifts by $\frac{1}{2}$ a period. For a big system, interactions with the environment will cause $|\psi_1\rangle$ and $|\psi_2\rangle$ to have uncertain energies (and more uncertain the larger the system) which will make them oscillate at uncertain phases and thus introduce uncertainty in their relative phase. Once this uncertainty reaches $\pi$, the interference pattern is washed out.
Another technical point that is important is that for these large systems it is typically only one of the system's degrees of freedom that can be placed in a cat state. Thus for the buckyballs you can see interference patterns for the centre-of-mass position, but the relative positions of the atoms in the ball will probably decohere quite quickly. (Of course, the same holds for the microwave resonator linked to above.)
If you look at it like this, then it does begin to look like one needs consciousness to collapse the wavepackets, and some people have even claimed that even consciousness doesn't. Whether this is right or not, or what other physics is behind what we perceive as paradoxes, well... find out and you'll get your Nobel.
