I know the answer. The problem is most cases is by No space left on Server for /tmp folders. But you can still upload files using FTP tool and File Manager from cPanel.
That's why you may have also that extra error in Joomla Module Manager
An error has occurred. 3 Error writing file '/tmp/' (Errcode: 28 - No space left on device)
CHECKINGS:
In cPanel (for domain), go to the Server Information > Service Information, and find those lines (at the bottom):
- Disk /tmp (/tmp) - 100%
- Disk /var/tmp (/var/tmp) - 100%
If you have 100% or 99% value there it means that Joomla/WordPress cannot upload ("big") files anymore.
SOLUTION:
If you’re on shared hosting and cannot clear /tmp yourself, contact your hosting provider and ask them to free up space or increase the /tmp allocation. Those folders must be almost empty (25% or less).
Later you will be able to upload files using Joomla Media, SPPB Media and use Modules. As you see there is no relation with used Joomla version, Template nor Component. Yes, I had the same problem on Joomla 5, and I solved it.