I think that it is amazing to think that any piece of cloth has endured 1000, 3000, or 6000 years The trick is not to believe everything we think. ;o)
Research often turns what we think upside down.
For example:
Here's a piece of textile from Peru that dates back to Jesus era = It's 2 ft x 6" = a border fringe...
"Border fragment Paracas culture,
Peru 0 - 100 A.D. 
The Shroud is linen, a cloth known for it's longevity - and the dry, arid climates in which these textiles were, added to their ability to last...in addition, the Shroud has been carefully carded for for hundreds of years - keeping most air and weather well from it.
How about some 8,000 year old cloth...in Florida.
Bye, Bye Beringia
""This cloth will set the example," Doran said. It is rare that fabric textiles even 1,000 years old are preserved in the United States."
*All told, 87 cloth fragments from an estimated 67 complete items were recovered from the dig. The cloth was made from the leaves of sabal palm. The pieces reveal five different methods of fabric making, all without benefit of a loom. Even so, some fabrics are woven as tightly as a cotton T-shirt. Others are made more loosely twined into blankets, capes, and toga-like garments.