Animals can handle a certain amount of in breeding. The fancy guppy is a good example. I bred them for 2.5 years. Eventually, your inbred bloodline becomes weak. After several generations you will start to notice undesirable characteristics.Like the fancy guppy, the BC's will pass on undesirable traits. Make sure you start with a healthy stock, preferably captive bred.
There are remote tribes of native people, and even religious groups here in the US, that are only allowed to marry individuals from the immediate community. This practice has been going on so long that people from these cultures are regularly born with extra fingers or toes. This comes from a "stagnant" genetic pool.
Both captive and wild animals often have no other choice. Our world is everchanging. Genetic diversity plays a major role in keeping species, as a whole, alive.