My grandmother and her sisters all used sheets for backings. One of her sisters is still an avid quilter and even uses sheets in her tops if she happens to find the perfect color. They all were and are hand quilters. Usually, the cheaper, lower thread count sheets work best. The cheaper sheets at Walmart use to be 100% cotton, but I think most of them now have some polyester in them as well. Target has 100% cotton sheets and I found some a couple years ago that were on clearance during a changeover from spring/summer colors to fall/winter colors and they were only $3 -$4 each. My mom bought all she could from the Target where she lives and I bought all I could here. I quilt on a longarm and often use sheets as backing for my "everyday" quilts. I do also have a collection of backings that we have bought on sale and use those as well. I haven't really seen any difference in the way they quilt up.
Oh, and those quilts that my grandmother made...the sheet backings have held up as well if not better than the tops. Sheets are made to be heavily laundered and most last for years with weekly washings. Most people, however, don't wash their quilts weekly.