I just set up a new iPad with a clean install and logged into iCloud and tapped 'Use iCloud' when starting iBooks. When I try to download a previously downloaded book from the 'My Books' tab in iBooks an error message with the following appears:
Couldn't Connect. iBooks couldn't connect to the iBooks Store. Check your Internet connection, then try again.
I tried deleting iBooks and not tapping 'Use iCloud' when starting iBooks but I get the same error message. The "Featured" tab opens just fine so it can connect to the iBooks and the App Store app works just fine. I also tried using my phone as a hotspot to make sure it wasn't an issue with the wifi I was on.