I've made domestic BD award bookings within a couple of days of the flight on two occasions.

More interestingly, I managed to obtain a supervisor's permission last summer to make a US-UK booking less than 48 hours in advance, when I needed to get home quickly to handle a family emergency.

So they can do it, if they want to. However, as it probably involves a bit of extra effort on their part, the people in the ICC aren't keen to make an exception if they can avoid it.