Yes. I have done this...twice.
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When I was pregnant with my first, DH loved the name Maximilian. I had a dog named Maximilian. Not just Max, like a normal person would do with a dog - my dog's name was the long version. I eventually conceded and we used the name. We call him Max. It's surprisingly uncommon in this area although I think it's generally a more popular name...I've never run into another Max. Just a sidenote: our child with the most unusual name - something that doesn't even rank on the Social Security Administration's list of top 1,000 names for the past 50 Â years - is the one that we run into a lot. There are two other girls in our church with the exact same name (first and middle).Â
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Back on topic...our youngest is named Veronica. At the time I didn't know it but DH had a childhood dog named Veronica. Now we have two kids with names that were once used for one of our pets.Â
 For what it's worth, I never look at Max and think "black lab" and DH doesn't look at Veronica and think "pug." They are definitely their own person.
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Granted, it's our own fault for giving People Names to pets. ;) I will keep this in mind if we ever get a dog, because in the past almost all of my pets have had very distinct people names - Peter, Donovan, Maximilian, Frederick...