Just to update and resolve this thread.
Thanks to Debra’s tip, I tried switching off my Sucuri Sucurity plugin but the problem persisted. (Which is good in a way as I didn’t want to lose the Sucuri plugin)
Johan managed to figure it out after examining my exported post snippets file – basically, if you have more than 100 Snippets saved, you might run into my issue which is that you cannot save any further Snippets to the plugin.
The solution is to go to your php.ini file and change the max_input_vars value from 1000 to 2000 (which will then let you save 200 snippets). If you aren’t comfortable with editing stuff from the command line, you will need to ask your hosting company to do it. (May be an issue for shared hosting).
This fixes the problem instantly. Thanks Johan!