I have an iPad 1 with iOS 5.1.1 that won't download any apps or update its OS. It was donated to my school so it was totally wiped before we got it.
I've tried the work around previously mentioned of downloading the apps in iTunes then dragging them over to my device that is also connected to iTunes. When I do this, a dialogue box comes up saying it was unable to install the app because the app requires a newer version of iOS. I'm really hoping Apple didn't make it so a perfectly good piece of tech is obsolete by not making iOS upgrades or apps compatible with their older technology.