In the investigation of index formulas for non-elliptic Fredholm operators, interesting new phenomena appear that are not apparent in elliptic theory. In testing the boundaries of known approaches to index problems, fundamental problems arise when we try to extend the theory to more general Fredholm operators. I want to discuss some "negative results" that may be of philosophical interest, partly in the hope that a clear exposition of some of these issues might elicit new ideas from the audience.