Only issue with bears in the area has been getting into improperly stored food. Never heard of a mountain lion attack in this part of the Olympic Peninsula, but there's a great picture in the Lost Resort of one standing on a boardwalk. Obviously the photographer survived ;-)
There are bears, and cougars in the area, but I wouldn't be overly concerned. A group of hikers had ran into a bear, and they said it seemed pretty used to people so it didn't really pay them a lot of attention.
There are some, but you don't have to be too worried if you follow all the appropriate safety precautions for food storage and if you see one leave them alone.
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