Tom Brady -- Deflategate Do-Over Request Rejected ... 4 Game Suspension Seems Unavoidable

Tom Brady's running out of options ... because a federal appeals court just shot down his request to get a rehearing in the Deflategate case -- meaning it's unlikely he'll get out of serving his 4 game suspension.

Brady had filed a request for a rehearing with the 2nd Circuit Court of Appeals -- but Wednesday, the court formally denied Brady's request.

The New England Patriots QB can still appeal to the U.S. Supreme Court -- but there's no guarantee the court would take the case ... and even if it did, it's unlikely the case would be decided before the 2016 NFL season begins.