Yes you can. For us it was Free as long as take your tent and follow safety rules.
I don't believe that you can camp at the actual Bridge to Nowhere; It's property owned by Bungee America. The trail on the way to it, however, might have some spots along the creeks that you can camp at (there were old fire pits scattered along them), although I don't know if it costs anything to do so. I recommend contacting Bungee America to ask them about both instances.
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