I very frequently encounter data which fit a normal distribution except for a few extra points in the tails. The QQ plot is basically straight in its middle section with the ends wandering away from a straight line. This is after removal of unambiguous outliers. The data are the yield of grain from small plots in series of agricultural field experiments. I know of no theoretical model which could explain this better than the normal, so I use this and advise against over-reliance on significance tests.