Yes, they do. My second government order was paid by an RBI cheque :) And, as usual, the employees of the Karnataka Tourism dept. had no clue if the cheque would cause any "issues". The ICICI Bank employees were looking at me suspiciously! This was in 2011. Apparently, nobody at the bank (Their Bangalore head office), had seen one.) So, they called people in Mumbai.
All branches of the government, pay vendors through RBI cheques.
Dont think other people could these cheques issued, though. Still not a billionaire :)