IP Networking

I have a TCPIP server application (a Vendor package) which by default allows 10 connections. It provides a parameter to allow us to increase the maximum allowable connections in case it is needed. Intermittently this application is failing with maximum number of connections reached even when there is no client activities. I am the application support and do not have root authority. When the error occour, I use netstat -an to display the connections. But I do not see any connection to the port that the server application is listening on. The server application is listening on port 7780, the command I used was netstat -an | grep 7780. But there no connection showing. I also pipe the output of netstat -an to a file and read through it carefully. I did not see any connection to that port. Is it possible that netstat -a will not show all connections or do I actually need root authority to see all ? If not, then how can I determine what is the problem.

Perhaps someone can help mr to under this lsof output
When the maximum connection 10 error occur I tried
$ lsof -i | grep 2773156
AppServ 2773156 apuser 263u IPv6 0xf100060005936398 0t0 TCP *:7780 (LISTEN)
:
AppServ 2773156 apuser 283u IPv6 0xf100060004feca90 0t0 TCP dev1.cs.ata:*->10.126.73.4:45366
AppServ 2773156 apuser 288u IPv6 0xf10006000287d290 0t0 TCP dev1.cs.ata:*->10.126.73.4:45936
:
AppServ 2773156 apuser 323u IPv4 0xf100060005563b98 0t0 TCP dev1.cs.ata:53553->dev2.cs.ctc:25550 (ESTABLISHED)

Where 2773156 is the application process and it is listening on 7780

I saw the 7780 listener and 9 of
AppServ 2773156 apuser 283u IPv6 0xf100060004feca90 0t0 TCP dev1.cs.ata:*->10.126.73.4:45366


I think this represents 9 connections. But it does not say what port. Instead it has :*.
What does :* means ? If this are connections, why there is no status (such as ESTABLISHED) like the one coming from dev2.cs.ctc:25550 ?