Here are some screenshots showing basic stuff related to solving network connection issues. Consult your peripheral/hardware manuals/webpages/support (console, modem, router, etc.), and your ISP and Microsoft customer support for more information if you are having problems.
Xbox.com web site offers information for people who have network issues. In this example we are visiting xbox.com > Support > System Setup > Xbox 360 Support > Resources > Troubleshooting Guides > Troubleshoot Your Xbox Live Connection.
Console manual in xbox.com > Support > System Setup > Xbox 360 Support > Resources > Instruction Manuals > Xbox 360 Console.
Xbox Live and xbox.com website had problems 2007-11-10 around 11 pm (Finnish time): I couldn't sign in to the xbox.com website, and in Xbox Live I was unable to send messages to friends, see their Gamerscore, etc.
No Gamerscore information for one of the Xbox Live gamertags 2007-11-10 11:47 pm (Finnish time).
Pinging xbox.com reveals msn.net had package loss problems (see Ping Test Program section below for more information) 2007-11-10 11:56 pm (Finnish time).
Xbox.com website was down for maintenance 2007-11-13.
If you have connection to Xbox Live but the connection is lagging or breaking a lot and if you also have a computer, you may get an estimate where the connection problems might be by using a program that can ping web servers. From the above Ping Plotter Freeware (Windows XP) example you can see that Microsoft's network might have too many users in the weekends because of the 28 % package loss between two msn.net servers (hops 9 and 10) when pinging www.xbox.com. For security and privacy reasons the IP address and DNS name for the first machine (my laptop) are hidden. (Avoid giving your console's/computer's IP address and DNS name and your real name, address, phone number, credit card information, etc. for example privately to strangers or publicly on the Internet).