A few reference points from elsewhere. Mostly related to what might be in a high rag content paper.
Oldest European wool textile - about 1500 BC. preserved in a Danish peat bog. (Nearly no oxygen environment)
Oldest linen - about 7000 BC !
A fragment wrapped around the handle of an antler tool. Preserved by partial fossilization from the calcium in the antler. There are fragments from Egypt that are about as old.
http://ounodesign.com/2008/10/30/the...-in-the-world/
Flax fiber fragments indicating use in textiles have been found in digs that are as far back as 34000 BC
Oldest cotton - 3000-2500 BC Pre Incan grave cloths
Oldest silk - About 1300 BC.
The oldest paper items found are Chinese and from right around 2000 years ago. They include flax and mulberry fibers.
So, assuming the T cards are on high rag content paper I'd guess that with a bit of care there might be a few left a thousand years from now.
Steve B