I have found a solution!!!!! (At least with the Iphone/ S3 combination)
Please pass this along to any other bulletin boards or people with Iphone.
It seems that the Iphone's settings have a lot to do with it.
After WEEKS of not getting my wife's texts from her IPhone, I changed my Galaxy S3's Sim card, have done powerdowns, soft and hard resets, I looked at my wife's Iphone - and did the following:
On her IPhone:
1. Do a hard reset (Press and hold top button on IPhone until it shuts down) then turn it on.
Upon powering up, Hers said something like "add text phone number" when it powered up. I opened the error message box, and even though it showed the correct number, I went ahead and deleted the last digit on that screen, and keyed it back in, and saved that as her text phone number. (Im not sure what effect that had, but I did it anyway).
But then I noticed: a. Her IMessage was set to "ON" which I left alone. But...
b. Her "Send as SMS" setting was turned "OFF". Which I turned to "ON".
Immediately her text messages started popping up on my Galaxy S3.
I'm not sure if the changing the phone number for texts and re-saving it fixed the issue, or the "Send as SMS" did it, but so far, the texts are coming through.
I'm not sure if it's a cure-all for all of the "Fail to receive text" problem, but it seems to be one solution.
NOTE: In spite of what the bulletin boards on the internet say, It is not a problem unique to any one carrier. It seems to be an IPHONE settings issue with the way it sends texts.