HI there Rillatarcool,
It sounds like you are getting a prompt that your storage is full but you have confirmed that it is not. It may be referring to your iCloud storage, in which case I would take a look at this article named:
If this is not the issue, I would quit all the running apps on your phone:
Force an app to close in iOS
1. Press the Home button two times quickly. You'll see small previews of your recently used apps.
2. Swipe left to find the app you want to close.
3. Swipe up on the app's preview to close it.
When you have done that force restart the device with the Home and the Sleep/Wake buttons and test the issue again:
iOS: Turning off and on (restarting) and resetting
I know you don't want to have to, but If the issue persists, the next thing I would have to recommend is to backup your device to iTunes, then restore it as new and verify that it reports correctly.
How to erase your iOS device and then set it up as a new device or restore it from backups
If it does, then restore your backup to either verify it still works and the software just needed reinstalled, or isolate the issue to the backup file itself. Thank you for using Apple Support Communities.