Did you get that error on all your LG's? Does that error occur only with a high user load or at least when you reach an amount of concurrent users? In case check if the LG's were overloaded by looking at the CPU/Memory/Network statistics.

Since the issue is a network issue probably caused by a network device (including your application server) the best is to use one LG which will have a different network path to reach your application. That allows you to eliminate some network devices (proxy, Load balancer etc...). The goal is to have that LG to reach your application itself without any LB or proxy. If you still have errors from that LG then you know that the issue is load that can't be handled by your application server.