iMessage not downloading pictures

I read a question recently about someone having trouble downloading pictures on iMessage. Instead of getting the image, it gives a "tap to download" icon (circle with the down arrow). When you click the icon, it does not even register that you have done anything. The answer they were given is that it was a storage or network issue. Well, I just got a new iPhone, and I now have over 200GB free. My iCloud storage has over 1TB free. I am connected to the internet on a fast network. So why am I still having this issue?