I feel like I rarely get upgraded on FHR (or really any of them TBH, except the ones that offer upgrade at booking), but there are many factors at play. I get there impression there are some factors which make it much less likely:

• Longer stays (might not be available for the entire span).
• Two beds. At the Langham Chicago I booked a room with two beds, and at check-in they honored the FHR upgrade by adding Club lounge access rather than a different view or higher room category. On a FSPP booking my parents had a double / double booking and they said they could upgrade the view if they were okay switching to a king.
• Unique room category or feature: for example when I book a "pool suite" at the Andaz Maui, it's a no-view suite with a little pool shared between 3 adjacent suites. One category up is a "partial view" suite is less desirable IMO, but they often won't consider it automatically if the upgrade isn't uniformly better (view change, special feature change, etc.).
• Booking a premium view room. For example, I'm currently staying at the Gritti Palace on an FHR booking and I booked a Landmark grand canal view room. At check-in they specifically mentioned a one category FHR upgrade could move me to a Sestiere (most are junior suites) with no view, but most people want the canal view. You'd need a 4 category upgrade to go to a suite with a canal view.
• Going to a Marriott or Hyatt of Hilton hotel. There are a lot of elites that can do confirmed upgrades or have upgrade priority over FHR / Virtuoso.

That's in addition to all of the usual things about high season, holidays, events, weekends, etc.