You can open thousands of connections with just 2 machines. Remember, a TCP connection consists of 4 items, a Source IP:Port connecting to a destination IP:Port e.g. 192.168.2.1:24543 -> 192.168.2.101:80. The TCP/IP guide is a great resource here.
On the destination side, it is often the case that whatever is listening is able to handle multiple connections on one destination IP:Port e.g. httpd can handle thousands of requests on port 80.
So with the above information it's easy to see that thousands of connections can be made from a single source IP address to a single destination IP address, all you need to do is create new connections, each using a unique port on the source IP.
How you do that is probably OS and tool specific and as you don't mention either ...