While 10k connections from one machine is certainly larger than you see in daily use, we use EC2 to generate load with our software as well and we on occasion run ~5,000 users from a single machine (and each simulated user will open multiple connections to the server).
More likely, you are simply overloading the network interface. You need to be sure the testing software you are using is robust enough to handle this situation. For example, with a test this large, our product will run the test controller on one machine and the load generators on separate machines. The controller is very tolerant of delays when communicating with the load engine since it is expected that network conditions may degrade at any time. If OpenSTA is not designed to handle this gracefully, you may run into this. If it is using all of the network bandwidth to hammer the server, there may be nothing left for it to communicate with you :(
I suggest using multiple machines to generate the load so that each one has enough bandwidth left for you to maintain connectivity with it.