This usually happens if the auto-save or one of the saves in your folder corrupts. It "hides* the rest of your saves although they are still fine.
First think you should do is make a backup of your user files: http://sims4.crinrict.com/en/2014/09/backups/
After that, just move the slot_00000001.save to the desktop and check if the game then returns.as playable.
If that doesn't work, try the same with slot_00000002.save until you find the guilty party.
The saves called slot_0000000x.save.verX are automatic backups for your games and they are not checked at the time of loading so you can ignore those for now.
If you need help figuring out which one is the guilty one, you can zip your entire saves folder and upload to dropbox, mediafire, googledrive or some other file sharing site and PM me the link. I can take a look if I can save your savegame. Even if you can figure it out, it would be great if you could share your save for the devs to squash this issue.