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.
Does anybody else have problems with mails not being delivered to their phones until hours or days after they've arrived in the inbox?