Got that discussion with my friend about inbreeding in fish. I dont wont to go into the jargon of things, but bottom line is that since the process of fertilization is not the same with mammals the recessive genes that would cause disease/skeletal problems/immune system deficiency and other 'beautiful' problems are been kept in check. They dont express themselves in the first 3 or 4 generations, but they are formed and are been carried around by the fish. After that the danger of expression is high and thereby the fish can suffer from problems in various areas should inbreeding continues. In essence its DNA becomes a pandoras box. You dont know what will come out.
This studies were conducted in various salt water fish and certainly not in bettas though.