I have a Droid Maxx. I get a notification that I have a text message. I open the Messaging app and I see "Downloading". Sometimes the text will finally appear but it can take from several minutes to several hours. This happens from a variety of contacts, both Android and non-Android. It usually happens when the text contains a photo. It even happens when I send a text to myself. I have seen at least 3 other posts on with this same problem but I have never seen a solution. My phone was hard reset at the Verizon store a week ago and the problem still happens.
Currently I am at home, on wifi, 72 Mbps. It does not seem to matter if I am home on wifi, or away on 4G with full bars.
Also, I misspoke ~ I have a Droid Razr M (not a Maxx)
You can delete the cache, and data within your messaging service app. Then restart your device and see if that helps.
If you have message threads you want to keep, you'll want to back those up before you do this though.
You should be able to find your messaging app under settings>application manager or some variation.