It seems to be turning over fast enough. When you are turning it over and it does not start does it smell like raw diesel? How many hours on the engine? Next thing I would do is a compression check. Was the oil milky brown or black? It seems like they auctioned off a tractor they knew had a issue. The T503 has a nice engine it the CAT/Perkins. How warm is it there. I know some diesels have issues and need glow plugs even at warm temps.