What does NAT mean to me and my friends?
Suppose you want to host an Xbox Live game and you're using a router that's classified as open. As the following table shows, you can be confident that your router's NAT type will not keep your friends (who may also be using routers) from finding your session, hearing your voice, and enjoying a great game.
But let's say you're using a router that's a strict NAT. When you want to host a game, you now have something to consider. Anyone else who's behind a router that uses a moderate or strict NAT may not be able to connect to your session.
Let's look at one final example going the other direction: you joining someone else's game. Will you be able to participate fully with everyone in the game? If you have a moderate NAT on your home network, you can get an idea of what may happen by consulting the table. If the players you want to play with are using strict NATs, you will likely experience problems. You may not be able to hear them and they may not be able to hear you. Your Xbox 360 console may not be able to "see" those players at all.
Anyway, as you can see from the chart, you should be able to connect, but if you cannot, then restricting your NAT from open to anything less will not improve the situation. There are likely other factors at play that are preventing you from connecting to your friends. Some people have even reported not being able to connect when all parties involved are set to open.