I've used sheet for quilt backings and even for the front of an applique quilt. I haven't had any problems. I've found that muslin sheets are too rough for me to use, so I used to use percale sheets (210 thread count). Any lower thread count sheets should work (100% cotton). So that would be up to 300 thread count, but some people report that 400 thread count sheets work as well.
My first two quilts were made with some poly-cotton blends left over from sewing clothes. I finished the quilts about 40 years ago. I'm finding that the blended fabric is quickest to wear out, so I've been replacing those blocks in the quilt as they wear out. I would suspect that the sheet used as backing of your quilt that wore out was a poly-cotton blend. Just get another back for your quilt and it should be fine.