It turns out that another process had been randomly assigned the ports that Networker had to listen on, to an outgoing TCP connection. Which, of course, meant that Networker couldn't bind to those ports to LISTEN. This is the first time this has happened. But it's a potential time bomb for any service that listens on specific ports. Such as Oracle, Weblogic, SAP, etc.
Linux controls what ports are randomly assigned using two sysctl's, ip_local_port_range and ip_local_reserved_ports. Unfortunately, the Oracle installer prerequisite check requires that ip_local_port_range be set wrong (1024-65500, which includes their own listener port) so we have to work with the other one, ip_local_reserved_ports. It's a "comma-separated list of ranges", so for us, I picked an excessive range for our big 3 applications- Oracle (1520 - 1530), SAP (3200 - 3699), and Networker (7937 - 8065).