Can't Connect Online

Steam

Make sure you have updated Steam and are running the latest version.

PlayStation

PS4

You need a PS+ subscription to access PlayStation's online multiplayer service. Without a PS+ subscription you cannot play online.
Contact your internet service provider to check the status of your internet connection.
If you cannot connect to PSN, check the status of PSN at the Sony website for your region.
If using wireless, test your connection when wired.

PS3

If you cannot connect to PSN, check the status of PSN at the Sony website for your region.
Turn your PS3 off and on, waiting for 30 seconds between. This gives the system time to refresh network settings.
If using wireless, test your connection when wired.
Remember, it is not possible to connect to the Internet via the USB connectors on the PlayStation 3system. The PlayStation 3system does not support USB wireless network devices. Make sure you are using an ethernet connection.
In some instances, it is necessary to restore the PlayStation 3 system’s default settings before establishing a network connection.
Before using with a PlayStation 3 system, an 802.11n router must be set to mixed mode, so that it supports 802.11b or 802.11g wireless signals.
Ensure that all network equipment is correctly configured and that all cables are functioning correctly (if in doubt, try using different cables).
Ensure that all network equipment firmware and drivers are functioning and up to date.
Check that the wireless access point (usually a router or hub) is functioning correctly by attempting to connect the PlayStation 3 system directly to the modem.
Contact your internet service provider to check the status of your internet connection.

PS Vita

Go to the home screen and select [Settings] > [Network] > [Internet Connection Test] to check your internet connection status.
If you cannot connect to PSN, check the status of PSN at the Sony website for your region.
Restart your Vita - turn it off then turn it back on. Yes, it's the good on/off trick! Sometimes when Vita is connected to the Wi-Fi for a long time, it automatically blocks the Access Point.
Turn off/on your router. Sometimes a simple restart solves it. Go to the Vita home screen and select [Settings] > [Network] > [Wi-Fi Settings] to connect to the access point again. You can also make sure that your router has the latest firmware.

Check it's not a DNS error. Ignore the router's DHCP settings and manually assign IP addresses, gateways and DNS servers to your device. This helps with dropped connections, IP address conflicts and failed DNS requests.
If you get a message like "Could not connect to the server or device within the time limit (NW-5603-4)" go to your WiFi settings in your Vita. Near the bottom of the list is an option to "disconnect Wi-Fi connection automatically". Tick that box.