It's very clearly stated in the Hilton Reservations FAQ:

https://secure.hilton.com/en/hi/res/faq.jhtml#07

"7. What ID do you need to have to get the government rate?
The Federal Government rate is for those employed by The Federal Government or the US Military. They will be required to show proper identification, Government issued ID or Military Travel Papers upon check-in at the hotel. Please note that the State Government rate for state employees is for the hotels in the state in which they are employed. State Government ID will be required at check-in. The Contractor rate is for any government contractor working on behalf of a Government Agency or the Military. Contractor ID will be required at check-in. Some hotels do not accept the Federal Government/State Government/Contractor rate, so check with the hotel directly."

I too reserve using the state rate offered at any property but I then follow-up with the property and talk with the revenue manager. I have yet to have one refuse me once I have booked an available room at that rate and provide them with my confirmation number.