FarSight have opted for a hard limit of three saved game points for each game. You will use those slots up if you adopt the backup strategy described in Maintaining Your Local Game Progress. You must delete older saved games to maintain a current save profile.

TPA Android users have confirmed that other games with this feature have more than three slots available, and clearly-defined Save and Load buttons on the Saved Game tile itself. This is an area of improvement FarSight can fix now, to improve the customer experience.

There is a gotcha in the way you delete the Saved Game files with the current GPS Saved Games UI. See How Do I Delete a Saved Game? to avoid losing data.