I noticed that my data skyrocketed this last couple weeks and I thought it might be due to the new Apple Music service, but I started thinking that maybe it was because cellular data was being used even when wifi was enabled. I decided that I would turn off Cellular Data today while I'm at work and wifi is readily available. My understanding is that pretty much anything that requires data should still work with wifi enabled and connected to a reliable wifi source. Then text messages started failing and Visual Voicemail wasn't available. I don't know if it's a direct correlation to turning off Cellular Data and that's how it's always been, but I've just never noticed before. When I turned Cellular Data back on I received several text messages that were sent more than an hour before I received them. One was an image from an Android user and the others were in a group message where 2 of the 3 participants were iPhone users and the 3rd an Android user. I don't know how to more accurately test this, but wanted to see if anyone else was noticing this or has any thoughts on it. I'll be filing a bug as well.
Cellular data being used instead of wifi
My girlfriend who is still on iOS 8 has been unable to send group messages without cellular data as well, so I don't think this is limited to iOS 9.
Did anything else fail except for text messages and Visual Voicemail? Those features seem to me to be heavily phone-oriented, so that's why they might require a cellular connection.