I am having this issue with Gmail, even though the Google account shows that gmail is selected to sync emails. Not sure this is an SIII issue though as my wife's Drod Incredible LTE is having the same issue. Spotty push notifications, where for instance I got a notification of 3 emails that are dated from yesterday but the notification didn't come till this morning. Obviously the Gmail update to allow notification bar replies and deleting does no good if you only get '4 new emails' once per day or every other day!