What's the maximum number of connections that I can expect before the system slows down considerably, or crashes?
That's impossible to say with any certainty.
You should run a load test to find out.
I don't think the limitation will be the network stack. It should be able to handle many, many millions of connections per day. The bottleneck will most likely be the application.
Assuming your application is a socket server with a custom protocol, you can write a simple, multithreaded Java client to connect to your server and run it on several PCs.
If you don't want to deal with writing a multithreaded client, you could use JMeter's JavaSampler functionality. You'd have to write a class to excercise the server and let JMeter do the multithreading and reporting.
If your application communicates with clients via HTTP, it's even simpler. Just use one of several load testing tools like JMeter or LoadRunner.
[ July 04, 2006: Message edited by: Scott Johnson ]