Answer: I would not expect the neighbor's cat to be the cause. Deer, rabbits, wood chucks, and slugs can all damage plants and may work at night or early in the morning when you would not see them.
The peony bloom problem could have been due to weather or a fungal disease, if it happens again youmight want to contact your county extension for a more specific diagnosis and suggestions as to what to do for it. In the meantime, make sure to remove all of the dead dried up foliage and stems left over from last year.
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