When you put in a save name it will also show you the path name at the top where it was stored. In my case it's C:/users/aaron/documents/Autonauts.

You can change this path to somewhere else if you want when you save.

This can happen if you try to save your level file in a place that your computer won't give you access to. An example of this would be anywhere in the "Program Files" on Windows. If you try saving to somewhere safe like the Desktop does that help fix the problem?