Sometimes when connection to Azure SQL DB failed you want to test the network layer to confirm this is not network issue that prevent you from accessing your Azure SQL DB.
Note that this is relevant only for connections originated out of Azure as described here: Ports beyond 1433 for ADO.NET 4.5
Using these tests result with other than success as shown here means you have network issue with accessing the server.
This test should succeed regardless of the firewall settings on the Azure SQL DB.
More information about troubleshooting connectivity can be found here: Troubleshooting Connectivity Common Scenarios
Using PowerShell (Recommended)
Use the following command to confirm you have true for the TcpTestSucceeded
Test-NetConnection <YourServerName>.database.windows.net -Port 1433
Download the PsPing tool.