Definitely! I do this a lot, and one my grandmothers did ALL of her quilts this way. If I use flannel on the back I almost always bring it over to the front as I like the slightly fuzzy edge.
The length of the leftover bit depends on how wide you want your binding to be, and if you need a 2nd fold for raveling - if you're using something that doesn't ravel you don't need a 2nd layer.
For flannel or cotton batting, I usually do as Tartan described - leave about an inch (I probably actually do more like an inch and a half because I like wider binding sometimes), fold the raw edge in to meet the batting edge, and then fold the whole shebang over and stitch it down. I don't press the folds but you could if you wanted to.
If you glue-baste ahead of time, that running ripple doesn't (usually) become a problem.