I have the same problem except that I am using a New iPad, iOS 5.1.1, Pages 1.6.1, and have a good Wi-Fi connection (not 4G).
When I try to copy from WebDAV or copy to WebDAV in either Pages, KeyNote, and Numbers, I get the same message, "Copying documents from WebDAV requires an Internet connection." I do not get the window which allows me to specify my WebDAV server as all of the information that I've found on the Internet says I should get.
I have "Use iCloud" turned on for these APPs, but I've also tried it with it off, but it made no difference. I am able to use the Dropbox APP, Google Drive APP and OverTheAir APP to get files and even upload pics and movies, but can't use the WebDAV feature of these iWork APPs, so I really can't see why this has anything to do with not being able to establish an Internet connection.
Not being able to copy from or to WebDAV sure is going to make sharing files less convenient (i.e. it seems e-mail is my only option to share documents from my iPad to at this point).
It seems this kind of support community is not providing much support since the previous person posted his message over a year ago and has not gotten any answers.