I'd also guess that you aren't using a correct password, or your router is blocking access from unknown devices (this is common, and relatively easy to fix).
If you have admin access to your router, go in and check the connected devices. If your iPad is telling you it's connected, it should show up in the list, and you should be able to go from there to get it whitelisted. You said you don't have it on, so other things can be causing it. Password mismatch for example, or other router features can block what certain devices have access to in some cases. If the iPad is showing up as at least trying to connect (this isn't as easy to ascertain, varies by router model, and too complex to explain here how to make absolute sure it's at least trying to connect) it isn't a hardware problem with the iPad, it's some software or middleware connectivity issue like firewalling (many routers do this), wireless being set up appropriately for some devices and not for others, the list could go on.
Check your router to see if it's there, and see if you can find any problems. If you have a higher-end router, it's software settings page will give you a lot of information that could point you in the right direction.
