Not able to open PGadmin

If you're experiencing a "connection timeout expired" error when trying to connect to your PostgreSQL 13 server using PgAdmin, there are several potential causes and solutions to consider:

1. Check your RDS instance's public accessibility: Ensure that your RDS instance is set to allow public access if you're connecting from outside the VPC.

2. Verify security group settings: Make sure the security group associated with your RDS instance has an inbound rule allowing traffic on port 5432 (default PostgreSQL port) from your IP address or the appropriate range.

3. Confirm subnet configuration: If your RDS instance is in a private subnet, ensure it has a route to an internet gateway if you're trying to access it from the internet. Alternatively, you may need to use a bastion host to connect.

4. Double-check connection details: Verify that you're using the correct endpoint, port, database name, and credentials in PgAdmin.

5. Check for firewall issues: Ensure that your local machine's firewall is not blocking outbound connections on port 5432.

6. Verify Network ACLs: Check that the Network ACLs for your VPC are not blocking inbound or outbound traffic for your RDS instance.

7. SSL/TLS issues: If SSL is enabled, ensure you're using the correct SSL mode and have the proper certificate configured in PgAdmin.

8. Try a different PgAdmin version: In some cases, there might be compatibility issues with specific versions of PgAdmin.

If you've verified all these points and are still unable to connect, you may want to check the RDS logs for any specific error messages that could provide more insight into the connection issue.

Remember, it's generally not recommended to make your RDS instance publicly accessible for security reasons. Consider using AWS Systems Manager Session Manager's port forwarding feature or a bastion host for more secure access if possible.
  

Sources
Cannot connect RDS PostgreSQL DB instance to IntelliJ or PgAdmin4 | AWS re:Post
Postgresql 16.3 VERSION unable to connect using PgAdmin | AWS re:Post
Troubleshooting connections to your RDS for PostgreSQL instance - Amazon Relational Database Service