This sounds like a push/background data issue, where refreshing forces the phone to update. I haven't had this issue since my OG HTC EVO, but toggling Sync on

This sounds like a push/background data issue, where refreshing forces the phone to update. I haven't had this issue since my OG HTC EVO, but toggling Sync on an account basis in the app could help. There was also a catastrophic bug in both Gmail and Inbox about a month ago where emails wouldn't notify, might be worth an update if you haven't in a while.