Typically, Apple would stop signing for the previous versions of iOS firmware, but just keep signing the latest iOS. So once we upgrade to the latest version, such as the current 6.1.3, 6.1.4 or even iOS 7 beta, we can not downgrade back to the previous 6.1.2, 5.1.1 and other older versions. So when you want to downgrade your device back to those old firmware, Apple’s server will not authenticate the firmware, and the device will go back to a recovery mode, to force you restore the latest firmware.
So how to check which firmware is still being signed by Apple? You can simply check the firmware status through this website
If the Signing status is “Yes”, then it means that your device (iPhone, iPad, iPod touch and Apple TV) can be downgraded to this firmware via iTunes.
You can also click on “Beta Firmware” to view which beta version is available for downgrading, for example, now you can downgrade iOS 7 beta 6 to beta 5, but can not downgrade back to b4, b3 or earlier beta versions.
If you want to downgrade back to 6.1.3 from iOS 7, follow this tutorial.