Welcome to Android Central.
What version of Android are you using?
What messaging app are you using?
I'm assuming that you're saying photos are not auto downloading in the app so that when you open the message the photo is already there and instead you receive a link that allows you to download the photo within 24 hours of receipt, is this correct? There can be a few reasons for this behavior. In most messaging apps there is a setting for behavior while roaming which includes not downloading large MMS messages. This is to avoid extra charges from the carrier as some charge fees for data and calls if the customer is on another network's system. This way someone could wait until they are back on their home carrier's network before using a bunch of data and racking up fees.
Another reason could be a setting in the app again that, like the first, postpones downloads again to save the user from fees when they might have data limits. Sounds like the same setting but this one is a little different because it doesn't automatically download large MMS messages to help the user avoid data overages. Since data is sold by the Gig using big blocks at a time for messages could eat it up pretty fast. On most carriers however regular messages, SMS and MMS, are counted separately and usually get lumped together and do not count towards the data total. Look through the setting for something that sounds like, automatically download MMS messages. Depending on the app and the device the wording will vary.
Too many messages. This is the one that gets overlooked frequently. This is also an older issue that doesn't seem to happen too often on newer devices but on older devices or devices that run older OS versions this was quite the issue. With this one having double or even triple digit thousands of messages causes the app to more or less be overloaded. It wants to keep messages ready for instant viewing by the user but this takes up a lot of memory and storage and some apps just can't handle keeping everything straight. This drain on resources and the app capacity combine to cause issues like this where things don't or can't automatically download. This also causes sending issues where messages begin to stall on the way out, especially MMS messages. There really is only one cure for this and that is to reduce the number of messages being saved. They don't have be deleted totally, there are tons of backup apps that would allow them to be saved but removed from the app/device. So if you have 10 or 15 thousand messages reduce that number to restore function. I personally knew a person with almost 30 thousand messages this happened to. The bad thing for them was they could only delete around 100 messages at time.
Finally, if you're not roaming, the settings are all set to automatic and you don't have thousands of messages clogging the works there could be a bug in the app. It may have come from an update that didn't install correctly or that left something behind after an update or that just got tired, who knows, but clearing the cache and sometimes the data for the messaging app helps. Since I don't know what device you have or OS you're using I can't tell you the exact method to do this but typically the way to do this would be to open the device settings, Settings>Applications>(insert messaging app's name here)>Storage>Clear cache / Clear data. Neither of these should cause the loss of messages but clearing the data may cause you to have to enter log-in information and make any setting changes again. This is why I would suggest clearing the cache first and testing before clearing data. I would also recommend getting a free backup app at the Play Store and just backing up all of the messages should the absolute worst case happen. Better safe than sorry.
Let us know what you find and if any of this was helpful and we'll go from there.
Best wishes.
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