Thanks for your input everyone! I wanted to confirm that increasing the somaxconn also fixed the issue for me.
Kevin
$ cat /proc/sys/net/core/somaxconn 128
by default, which is right about where the problem starts. After
$ sudo sh -c 'echo 1000 >/proc/sys/net/core/somaxconn'
*and restarting the PG server*, I can do a lot more threads without a problem. Evidently, the server's socket's listen queue length is fixed at creation and adjusting the kernel limit won't immediately change it.
