Global attribute correlations

Using the normal setup for a variable product, no you could not do this.

You can set what values are available for an Attribute and that can be done on a global level. However, you cannot set what values are available for an attribute, based on the selection of another attribute globally. That would need to be done by setting up your variations correctly.

So what I mean by this is that if you have an Attribute for each material, you could set the colours that each of those have. For example:

Cotton : red, green, blue
Leather: black, brown, gray
Canvas : white, tan, gray

There would be no issue setting up three different attributes and assigning the values to each of those. However, if you use these attributes for your variations, it would mean that each would be a separate selection on the customer’s side. So instead of seeing a single drop-down to choose their fabric type, they’d see a drop-down for cotton with color selections, then a drop-down for leather, and canvas, etc.

If you are looking to save some time from manually working through setting up each product, I would recommend importing these products by CSV. That way you can set up all your products on a CSV, then edit them far more quickly, in bulk, and upload them to your site.