Hotels usually assign guests rooms the day before their arrival. I was able to get a confirmed upgrade once by calling the hotel (not the national reservations line) two days before I arrived. The reservationist was able to see capacity information for my stay and put me in a suite right away. I checked online right after the call and found the reservation had changed.

Your results will most certainly vary. It helps to have reason for your request like, "I'm trying to decided between two hotels" or "It's a special occasion", etc. Best to call during a slower time at the hotel when the agent can spend more time on your request.