Vimcasts.org runs on Radiant CMS, which has page caching enabled by default. My problem was too many app instances spinning up. I've limited the PassengerMaxPoolSize and PassengerMaxInstancesPerApp both to 2, which seems to have fixed it.
I appreciate this burst of traffic to help me get to the bottom of this.
I appreciate this burst of traffic to help me get to the bottom of this.