Check that the game (and/or launcher) is allowed through the firewall. On Windows the built-in Windows Firewall will open a large prompt window the first time a program's access is blocked (although you may have to use Alt+Tab ↹ or exit the game in order to see it) but other firewall programs may prompt differently or block unrecognised programs without sufficient notification.
LAN play functions listed in many newer games actually require internet access (either during the initial session launch or throughout the multiplayer session) rather than truly working over the local network. Games with this issue should have LAN play listed as unsupported with a note about this requirement. This may also be caused by the firewall blocking internet access.
GameSpy servers have been shut down for all games. Some games have official or community patches to connect to replacement servers (refer to the game's page for details, or see the GameSpy page for general workarounds).
The PunkBuster software may have to be updated manually for some games (refer to the game's page for details, or see the PunkBuster page for general workarounds).
Some games with local multiplayer support expect specific control configurations (e.g. player one on keyboard/mouse and other players on controllers). See the Network section on the game's page for further details.