I've wanted to try this for a long time and I finally took the plunge! I have successfully put fleece as the backing of a cotton quilt! Yes--it works!
I got this Star Wars panel at a garage sale a couple years ago, and it recently resurfaced from the inner-most chamber of my stash. It was calling to me to make a hurry-up blanket for my young Padawan son for his impending birthday.
Make sure your cottons are all prewashed and preshrunk. I added a colorful scrap border and a regular border. I continued the fire theme by freelancing some free-motion flames in the scrap border.
Using fleece on the back eliminates the need for batting, and it comes up to 60 inches wide, so I didn't even piece this back. Two yards of length was all I used. With a fleece back, you can get away with less quilting than with cottons too. I think the best results would be obtained with a high quality fleece. I used an anti-pill fleece.
I layed out the fleece without stretching it, and then layed out the top over it. I pin basted well, to reduce stretching that could occur during quilting. I think you could add a regular fabric binding, but for softness, I opted to do a fold-over binding. It worked really well.
I rotory cut the fleece edge 3/4 " larger than the quilt top. Fleece doesn't fray, so I think a straight edge would be fine, but I used the scolloped blade on my rotary cutter (it was a little challenging to use a wavy blade againt a straight ruler). Then, you just fold it over in place and sew it down. It would be essential to use a walking foot with fleece.
My fold-over binding corners are always less than stellar, but fleece is so forgiving, that you can easily push the corners into place. I sewed to within a couple of inches of the corner from both ways, snipped off the pointed corner, and then worked it into a suitable corner shape, and stitched it up. Easy!