Eh.... it can.
Any time you need to make a new save it has to be written, so if you fill up the card with other stuff, each game is writing it save on a smaller and smaller 'block' of free space. When the space runs out, it won't be able to (fully) write the save which can cause problems.
Also if a file needs to write something to the SD card for whatever reason and there's no space it can cause issues.

However, 1% of 4GB is ~40MB which should be enough space for most of your saves.

 

Thanks for the quick replies! Much appreciated!