cap,
Well, as I understand it, the clay particles by definition are cohesive, and the silt particles by definition are not, regardless of the diameter of the particles.
Think of it this way: the silt particles are just really, really fine (generally spherical, and electrostatically relatively inert) sand grains. The clay particles are those that have a high enough surface area to exhibit cohesion when sufficiently hydrated - up to the Liquid Limit.
The nominal particle sizes for clay and silt defined in the various classification systems are oversimplifications that tend to be accurate for the most part. Hoever, they should not be relied on as an absolute indicator of the relative composition of the soil - the cohesive or cohesionless behavior of the soil is a better measure of the relative abundance of the clay minerals.
So, depending on the coarser particle size distribution, it may take the presence of a larger or smaller amount of the clay minerals (whatever their actual diameter) within the soil matrix to raise the Plastic Limit above NP (i.e., induce the soil to behave cohesively overall).
Please note that even when the PI is small (but non-zero), the soil does behave cohesively. It's just that there isn't enough of a cohesive fraction present to act over a significant enough range to classify the soil overall as cohesive, except when it is within the very narrow range defined by the LL and PL. However, if the expected moisture condition is within this range, then the soil should probably be classified as cohesive for the purpose of the analysis.
Hopefully, that was fairly clear.
Jeff