DESCANT is pretty much correct. I'll go one step further: When fabric covers start to fall apart it is because the fabric is OLD and deteriorating, and needs to be replaced. Stitching is the usual weak link. If the stitching is cotton, it rots. If the stitching is polyester (plastic) it dies from exposure to UV Light.
Spending money on the old stuff is really a waste. Either way, it's time to buy a NEW cover.
NB