Verification Of What Ports Are Open

I have used a tool called Net Tools to scan ports on my various servers to determine ports are open in the firewall through my host.

For around a decade, this tool has been accurate to determine what ports are open/closed.

I have used this tool to accurately find issues with the firewall with current host (dedicated server). The network admin says that net tools should not be trusted and even though the software claims the ports are open, they are not open.

I explained that less than a month ago, this software reflected that these ports should closed were reflected accurately as closed by scanning the ports.

Partial results:

Code:
Address Scanned: xx.xx.xx.xx

Scan Started at 10/27/2007 2:08:00 PM
Scan Finished at 10/27/2007 2:18:03 PM

Ports Scanned: 1 To 4000

Total Ports Found Open: 3794

Current Ports Found Open:
Port: 00004
Port: 00005
Port: 00009
Port: 00026
Port: 00044
Port: 00076
Port: 00083
Port: 00088
...
What are the ABSOLUTE best tools/ways that I can confirm that the ports are truly not accessible?