Can you put a new SIM card in an old iPhone?

If you put an AT&T SIM Card in a new iPhone, can you put a SIM card from another network into the old (previously on contract) iPhone?

You need to unlock the old phone to any network. Your local phone/gadget/shop/market etc. Should be able to provide this service.

If your phone is unlocked you can use it on any network. Now iPhones use Nano-SIM's ( 5/5C/5S ) unlike older ones which use Micro-SIM's ( 4/4S ). If you don't have the correct SIM size you will need to get it "cut" to fit into the SIM tray whereas if you have a smaller SIM card you will need to get an Adaptor ( Placing the SIM in this will allow it to gain size to work in your iPhone ).

The old iPhone would still be locked to the network, so it would have to be unlocked.

Before swap your Sim card, keep in mind the following steps.
1. Make sure your iPhone is unlocked one.
2. Check whether it's your current Sim card is suitable for your iPhone
If your iPhone is locked other than your current carrier, you will need to unlock your iPhone. For iPhone unlocking you can approach your service provider or else try to buy an unlocking service from any one of the online unlocking service provider like http://www.wickedunlock.com/apple-iphone/rs16wp9/ Once unlock it, you will be able to swap your Sim card without having any problem.