We had a house near Matthews Beach, and when we moved in there were many rabbits in the neighborhood and in our yard. We have a terrier-mix mutt, and there weren't rabbits in our yard for long...
We have had the opposite experience: over the years, mama rabbits have had multiple litters in our backyard. We have two pit bull terriers, and they would chase the rabbits (never catching them), so mama rabbit knows there are dogs. My only theory is that the dogs are a known quantity, and the dogs keep the real predators out of the backyard.
My dog definitely caught rabbits. He even brought two kits into the house. He's a sweetheart in every other circumstance, but he loves toys that squeak when he mauls them.
That, or rabbits are just that dumb.