Looks like some moire effect, since it only shows up where you can see another patterned surface behind the first. You can try hiding every layer except one, see if the grid effect disappears on the remaining layer. If so, then it's simply the visibility of one layer behind the other - in the real world example, the chains are not all in perfect alignment, so this doesn't occur, but in the case of Pattern they are all perfectly aligned and repeating, so that there is a pattern made from seeing one pattern behind another.
You could try adding some displacement to the base geometry, sort of like ripples in the way they hang, which will then randomize things a little, and / or you could try moving things around inside the geometry to be patterned so that it isn't 100% regular.
