My wife is a teacher, and had been under that assumption too. All of a sudden had several properties that questioned and two that refused to accept the rate. I asked for clarification for Marriott and got a junk email reply. The two properties that refused and changed us to rack (not even lowest available online rate at the moment of change) later credited us back upon speaking with the GM, who remained by the policy that Teacher do not qualify. (Though my wife is considered an employee of Pinellas County School Board with payroll directly state funded).
Anyways, its rare that government is substantially lower than m11 etc, but on occasion will still use it. (not at the two properties referred to above).