Thank you for posting.
It may help to determine whether the clients can physically connect to the server. On one of the affected clients, trying using telnet from the command line or terminal:
telnet ServerNameOrIP 5003
This should tell you whether any network connection can be established to the server.
Each client machine is also capable of sharing files. Try to share files from some and see if any other clients can connect to them. Again, this will help determine whether the problem is with the network, server, or the individual clients.
If certain clients can't connect to anything, it could indicate in issue with the server.pem file located in the FileMaker Pro installation directory. Do these files exist on each client and appear to be about 6-8 KB in size? If this is not the case on some of the clients, it may be possible to copy the server.pem from the clients that can connect.
The issue is solved:
it turns out that the "server.pem" is needed in the directory with the Filemaker-executable. If it is not installed or removed from the directory you will see that there is a FM-server in the local network, but you will never be able to see and open it's shared databases. It does not make any difference which version of Filemaker Pro is used, I've tested it with 9v3 thru 11v2 both the normal Filemaker Pro as well as Filemaker Pro Advanced. I've tested it with FM-server 9.0.3 thru 11.0.2.
I find it very surprising that this file is needed since in none of the tested combinations SSL is used, nor is it configured, nor has it ever been configured. Interchanging any of the "server.pem"files from any Filemaker Pro version to any other still lets the databases from any deployment be shown in any version of Filemaker Pro??????
This makes it a bit difficult to understand why I had this problem in the first place since the installer obviously should have copied the file "server.pem" into the Filemaker Pro-directory regardless of the ntp-settings on the workstation.
The only differences between the PEM-files are that both the Private-Key and the Fingerprint for the Public-Key are different, but the Certificate itself and the Certificate-request are both identical in all these files for each of the Filemaker Pro versions? Again I am not using any secure connections to FM-server at all, since all these networks are private networks with decent firewalls.
Anyway, the main thing for me is that my issue is resolved.