osmc@osmc:~$ grep Kodi .kodi/temp/kodi.log | head -3
04:27:09 10.277564 T:1958364080 NOTICE: Starting Kodi (17.0-ALPHA1). Platform: Linux ARM (Thumb) 32-bit
04:27:09 10.277637 T:1958364080 NOTICE: Using Release Kodi x32 build (version for Raspberry Pi)
04:27:09 10.277714 T:1958364080 NOTICE: Kodi compiled Mar 18 2016 by GCC 4.9.2 for Linux ARM (Thumb) 32-bit version 3.16.7 (200711)
What is the brand and model of the SD card and where was it bought ?
It is quite possibly either faulty, or a counterfeit card.
Many counterfeit cards report more storage space than they really have (for example report 16GB but really only have 4GB of storage) so they work fine until that amount of space is used up then further writes to the card are discarded. Thus you are unable to make any further changes to the file system - the card reports to the OS that changes were written to disk but in reality they were not.