Originally Posted by aiharkness
For what it's worth, I don't see any difference between SMS from contacts (and spam sometimes) and legit SMS from web services like two-factor authentication codes.
I'm with tsac on the general idea that the answer is something on your end, maybe a setting on the device, or maybe something that has been enabled at the carrier. The parental controls was a good thought, and maybe cycling it on and off is worth a try. But I would also just feel out your carrier's customer service and see what ideas they have.
Putting your SIM in another device and then setting up one of these SMS messages from a website is also a good idea. Borrow a device for a few minutes. See what happens. If the messages makes it through, then that tells you to focus on the device. If the message doesn't get through, that lends weight to assuming the issue is with the carrier vice the device.
My bet, however, is that the issue is with the carrier. If you can test with another device, I would do that before calling the carrier. And when you call the carrier, tell them what you have done as far as troubleshooting.
I do not have another "device". I will contact the carrier (AT&T), although they are enormously difficult to get an intelligent answer from, and see if they know anything.