curl typically makes a TCP connection to the host as an initial part of its network transfer. This TCP connection can fail or be slow, if there are shaky network conditions or faulty remote servers.
To reduce the impact on your scripts or other use, you can set the maximum
time in seconds which curl allows for the connection attempt. With
--connect-timeout you tell curl the maximum time to allow for connecting,
and if curl has not connected in that time it returns a failure.
The connection timeout only limits the time curl is allowed to spend up until the moment it connects, so once the TCP connection has been established it can take longer time. See the Timeouts section for more on generic curl timeouts.
If you specify a low timeout, you effectively disable curl's ability to connect to remote servers, slow servers or servers you access over unreliable networks.
The connection timeout can be specified as a decimal value for sub-second precision. For example, to allow 2781 milliseconds to connect:
curl --connect-timeout 2.781 https://example.com/