I don't think the answer that was given before is valid. I did some research and there doesn't appear to be a reason that TcpClient has any strict limit imposed. The other answer suggested a limit of 10 active TCP connections in Windows XP for a server service, but this is only true for specific application protocols (such as sharing, mail) provided by Microsoft.
I did find though that Microsoft added some restrictions in XP SP2 and continued through Vista SP1 to limit half-open TCP connections to 10. They also limited the number of new connections to 10 within a single second. These were both attempts to reduce the impact of virus software. Apparently these have not been very successful so Microsoft has finally decided to remove these in Vista SP2 and later.
This may have been the OP's actual issue - perhaps you are trying to create those connections too quickly in the client?
As I mentioned in my comments on the previously accepted answer, I personally write client/server software and have not had problems with a limit of 10. When I do tests, using Windows XP in the past and now with Windows 7 (the application limit is now at 20 connections), I am able to easily exceed this limit. I just retested and was able to create >140 active/established connections between one host (Windows 7 Home Premium) and another host (Windows 7 Enterprise); both client operating systems. This is using Java but there really should be no difference with .NET (which I've also tested in the past).