iPad 3 not connecting to a cellular network at all
Which iOS version is installed? Did it ever work since you bought it? Have you updated the iOS since you own the iPad?

Your description reminds me of my fahter's 2nd hand iPhone 3GS. He bought it and updated to the latest iOS 6.0.1 without knowing that the iPhone he bought was jailbroken. The update made the iPhone totally worthless because it could not connect to mobile network nor WLAN any more.