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Most animals in captivity have inbreed problems.
Budgie parakeets for example, there is only one color of them in the wild - green. Basically all the other colors we see are the result of selective breeding. In other words, inbreeding.
Inbreeding might not cause defects, but in theory it can certainly produce weaker offspring which can easily catch diseases and become sick.
In order to breed a certain trait, a lot of people are still using selective breed to produce the results they want. So yes, inbreeding happens a lot for pet birds and many other pets as well.
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