That's a great fundamental question. The answer is not fully known. In the basic form of quantum theory as we know it, the interference persists no matter how far apart the slits are and no matter what the properties are of the object passing through them. Nevertheless, there are severe practical limits and may also be some basic limits.
In practice, there are practical limits to our ability to see these interference effects. They tend to be lost when there can be excitations of internal modes of the object, such as vibrations in a C60 buckyball. You can picture the C60 wave going through the two slits as having decreasing probability over time of being in the same internal state. Only components in the same internal state interfere. In principle, cooling the C60's etc. enough forces them into the lowest energy internal state, but this procedure becomes more and more impractical for larger objects.
Even for objects with few internal degrees of freedom (e.g. neutrons) decoherence (loss of interference) occurs due to coupling to a complicated outside world. Every particle interacts some with its neighbors, by gravity if not by anything stronger. As the paths become more separated, the resulting states of the neighboring particles become more distinct depending on the particle's path. This too causes decoherence.
There are very active efforts to find ways to reduce such decoherence effects for various quantum systems, since decoherence limits the ability of a system to function as a quantum computer.
No one knows for sure whether at some point there will be a fundamental source of decoherence, beyond the various practical sources. That doesn't stop people from having very strong opinions on the subject, with varying degrees of justification. I'm just now struggling through a paper by Lenny Susskind, in which he proposes that fundamental decoherence arises from interactions with things propagating out to cosmological horizons, at which information is irreversibly lost.
(published on 09/04/11)