Yes. I have done this...twice.
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 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).
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. 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.
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...