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I'm not an Apple aficionado so this may come across as silly by some.
My wife's iPhone can receive text messages from anyone, at anytime, whether she is connected via mobile or wifi. However, when I try to send her a text from her iPad, they are not received on her iPhone until mobile is turned on. While I've tried several turn-this-on-and-turn-this-off scenarios, there are still a few more I could do to narrow it down a bit further.
I've tried Googling, but without really knowing what is going on (e.g., feature or bug), I don't really know what terms to search for.
"One man's wage rise is another man's price increase." - Harold Wilson
"Fireproof doesn't mean the fire will never come. It means when the fire comes that you will be able to withstand it." - Michael Simmons
"Show me a community that obeys the Ten Commandments and I'll show you a less crowded prison system." - Anonymous
If your actions inspire others to dream more, learn more, do more and become more, you are a leader.-John Q. Adams You must accept one of two basic premises: Either we are alone in the universe, or we are not alone in the universe. And either way, the implications are staggering.-Wernher von Braun Only two things are infinite, the universe and human stupidity, and I'm not sure about the former.-Albert Einstein
Without some subscription and or tool, you cannot receive text messages while it connected to a phone network.
You can be connected to wifi and phone, but text messages are sent via your phone company's network that knows nothing about your Internet network via wifi - unless you have separately subscribed to some service to sent text messages tot the one via the Internet.