Welcome to Android Central! Are these contacts saved to your Google account, or only locally on the Phone account? Do you mean that the contact names don't show up on the incoming call screen, or that the names disappear from the actual entry in the Contacts app?
Are you assigning custom ringtones, or one of the preinstalled ringtones? If they're custom, are they saved to Internal Storage or the microSD card? Make sure they're saved to the Ringtones directory of Internal Storage.