In my catalog, I have one non-default category: Fabrics.
You’ll see in the attached screenshot that all 15 of my SKUs appear under this category.
Per product, however, I get a Product Saving Error when I attempt to check off the category. For all but three of the products, the category is not checked. For three of them it is. The 15 products in the category screenshot are the full extent of the product catalog in the database.
Click thumbnail to see full-size image