Sounds to me like it may actually have been a fiberglass fabric that looked like cotton. I had this experience many years ago and those little fibers are actually bits of fiberglass. I remember that even the next few loads of wash had some in it. If the clothes are worn, they will be quite itchy against the skin. Another problem can be that some small fibers can become airborne and be inadvertently breathed in. That is why they have special people now to remove old fiberglass insulation and wear protective clothing, breathing aparatus and have a contained removal system.