First, please make sure you have at least the following versions of these apps installed on your phone (you can find these apps listed in Settings >> Application Manager):
- Google Play Games version 3.5.17
- Google Play Services version 8.4.87
If you have the these (or more current) app versions and are still experiencing issues, please try clearing the cache data for Google Play Services and see if that helps (you can do this by going to Settings >> Application Manager >> Google Play Services).
If neither of those things work, please follow the directions on this FAQ. A player told us that this helped her to get sign-in working again, even though she was ultimately reinstalling the same version that she uninstalled (just without all of the intermediate upgrades).