IOS – Issue with FaceTime and iMessage (iPod touch 4)

My sister bought an iPod touch yesterday. Everything seems to work fine except for FaceTime and iMessage.
In both applications, after she signs in with her Apple ID, she can see the next screen for about a second before it automatically reverts back to the 'Apple ID sign in' screen. No error message is shown.

While we couldn't find any information on our specific problem online, we did find some general iPod touch/FaceTime troubleshooting tips which we followed:

• We've updated to the latest version of iOS and iTunes
• We've verified her Apple ID via the Apple website
• We restarted her iPod touch
• We made sure there are no active restrictions under 'Settings > General > Restrictions'
• We reconnected to our WiFi. We also restarted our router just to be sure

I also tried logging into FaceTime with my own Apple ID on her iPod touch. Surprisingly, the screen no longer reverted back to the previous screen automatically, but I wasn't able to move on to the next one because my account wasn't verified.
After verifying my account on the Apple website, the problem occured on my account as well.

I love Apple products, I do. But after struggling with this problem for an entire afternoon, I want to strangle the guy that thought it was a good idea not to show an error code or at least a short description explaining the problem. On any other device I would've been able to figure out what exactly I was doing wrong in seconds, yet here they leave me guessing.