How a connection works in a nutshell
A connection between a program and a database server relies on a handshake.
What this means is that when a connection is opened then the thread establishing the connection will send network packets to the database server. This thread will then pause until either network packets about this connection are received from the database server or when the connection timeout expires.
The connection timeout
The connection timeout is measured in seconds from the point the connection is opened.
When the timeout expires then the thread will continue, but it will do so having reported a connection failure.
If there is no value specified for connection timeout in the connection string then the default value is 30.
A value greater than zero means how many seconds before it gives up e.g. a value of 10 means to wait 10 seconds.
A value of 0 means to never give up waiting for the connection
Note: A value of 0 is not advised since it is possible for either the connection request packets or the server response packets to get lost. Will you seriously be prepared to wait even a day for a response that may never come?
What should I set my Connection Timeout value to?
This setting should depend on the speed of your network and how long you are prepared to allow a thread to wait for a response.
As an example, on a task that repeats hourly during the day, I know my network has always responded within one second so I set the connection timeout to a value of 2 just to be safe. I will then try again three times before giving up and either raising a support ticket or escalating a similar existing support ticket.
Test your own network speed and consider what to do when a connection fails as a one off, and also when it fails repeatedly and sporadically.