No you cannot.

You may have seen trains like "CSX" along the track loaded with cars. So this may be why you have you ask yourself if you can ship a car by train. 

CSX and nearly all trains moving cars are used for vehicle distribution therefore used in bulk by large dealers and manufacturers. They do not ship personal cars. The only one left is Amtrak and they only cover the eastern seaboard. They also require you to ride on the train. It is not a door to door option.