To start with, no deviations from inverse-square behaviour have yet been detected. At the very smallest scales, it’s exceedingly difficult to detect gravity at all, much less measure it. It is, however, sometimes hypothesised that there might be deviations at some scales, and experiments are sometimes designed to detect these deviations (only sometimes, because sometimes the theoretically-predicted deviations are far beyond the capabilities of current experiments).
The most commonly-discussed models which predict deviations from inverse-square behaviour at small distances are those involving extra dimensions. In such models, space has extra dimensions, which wrap in on themselves at scales possibly as large as a tenth of a millimeter. However, only gravity can “feel” these extra dimensions, not any of the other particles or forces. The effect is that at small scales (smaller than the size of the extra dimensions), gravity would spread out in 3+n spatial dimensions, and therefore fall off as 1/r[sup]n+2[/sup]. At larger scales, however, due to these extra dimensions being rolled up, the gravitational force would appear as our normal 1/r[sup]2[/sup] (it’s already spread out as much as it can in the extra dimensions). And anything other than gravity would always behave as if there were three spatial dimensions. This model does not, however, say much of interest about the gravitons themselves, just about the spacetime they propagate through.
There is another set of models where gravity might deviate from inverse-square behaviour, if gravitons are not massless as currently believed, but rather have a small but finite mass. I suppose one might fairly call such gravitons “fat”. But in this case, the deviations would show up not on small scales, but on very large scales. The basic problem is that massive particles can decay, and generally do, after a time related to their mass. So if you have gravitons travelling between two very distant masses, not all of them will make it all the way (this is a vast oversimplification, but gets the idea across). So the gravitational force at very large distances will be somewhat less than inverse-square. But at short distances, most of the gravitons won’t have a chance to decay, so the effect won’t be noticeable.
And I, too, have heard rumors about the Eöt-Wash group detecting a deviation, but so far, that’s all they are, rumors. They haven’t yet said anything of the sort in their publications. The experiments they’re performing are very delicate, and it’s quite possible that the effect they’re seeing (if any; again, this is only rumor) are just experimental error. Naturally, if they do see a deviation, they’ll want to be dead certain that their results are solid before publishing. It would be exceedingly cool if there really were a deviation, but let’s not get our hopes up yet.