When you get anything that says your network connection timed out it either means you lost internet connection or there is something blocking the connection between the computer and the servers that it is trying to connect to.
If you have a solid internet connection and can browse to websites fine it's either an anti-virus program or a firewall on the computer OR a firewall in the router's settings that you use to connect to the internet.
Reconfigure any security software to allow Apple's servers to pass.
Useful information for configuring your security software:
IMPORTANT: iTunes must be allowed to contact Apple using the following ports and servers:
- port 80
- port 443
- phobos.apple.com, deimos3.apple.com, albert.apple.com, gs.apple.com, itunes.apple.com, ax.itunes.apple.com
For a more complete list of ports used by Apple products see "Well known" TCP and UDP ports used by Apple software products.
iTunes also contacts VeriSign servers during an iPhone restore and activation:
For more information see Update and restore alert messages on iPhone and iPod touch
Temporarily disable or uninstall your security software
If the current available version of your security software is incompatible with iTunes, you can disable the security software and test if it is the source of the issue. It might also be necessary to temporarily uninstall security software after making sure you can reinstall it and note any required licenses. A temporary deinstallation might be required because parts of some security software remain active in the background though they appear to be deactivated.
To reduce any security risks that security software is designed to prevent, consider making sure all files needed to reinstall it are available and disconnect network connections before uninstalling security software. Should this not be possible, such as if a download without active security software is required, that required step should be the only process completed without security software. For assistance in checking your configuration, disabling, or uninstalling your security software, please contact the manufacturer.
Once disabled or uninstalled, perform only the steps required to verify your issue has been resolved. Complete whatever update, restore, syncing, back up, activation, or other task you were troubleshooting. Once that task is complete, install and configure compatible security software to ensure your PC is protected.
If it isn't a program or software on the computer contact your Internet Service Provider and/or Router Company to configure or disable the firewall in the router settings.
To isolate the issue to this you can also try disconnecting the router and plugging your computer directly in to the modem but this does not help for troubleshooting time-out issues directly on any apple mobile devices only if you are syncing or going through iTunes on the computer.