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Yes, all animals have the potential to carry recessive genes that control undesirable traits. When breeding "close", more homologous pairs of genes are formed. Many deleterious traits are recessive, but so are many desirable traits. So, when the genes are doubled-up, we see the traits. We can select for them or against them. This means that by using inbreeding along with the right selection technique, you can select against the undesired traits and even eliminated them from the line. At the same time we are more likely to "fix" desirable traits.
With humans, we don't practice selective breeding, so we shouldn't procreate with relatives. With animals, we can test cross, out-cross, inbreed and line breed at will - and all with the same animals as one of the parents. The immune system will eventually become compromised with extended inbreeding, but no other undesired trait will remain in the line if you actively breed against it with the proper crosses. All breeds of domestic animals including strictly ornamental types, were created with a certain amount of inbreeding. Indiscriminate inbreeding is never a good thing, but purposeful inbreeding in the hands of the knowledgeable person can do amazing things.