Additional information for modern (systemd) linux systems.
There are many resources about this of values that may need tweaking (such as How to increase maximum number of JVM threads (Linux 64bit)); however a new limit is imposed by way of the systemd "TasksMax" limit which sets pids.max on the cgroup.
For login sessions the UserTasksMax default is 33% of the kernel limit pids_max (usually 12,288) and can be override in /etc/systemd/logind.conf.
For services the DefaultTasksMax default is 15% of the kernel limit pids_max (usually 4,915). You can override it for the service by setting TasksMax in "systemctl edit" or update DefaultTasksMax in /etc/systemd/system.conf