I've never tried particularly early, but when we've flown with our little dependent, purchasing tickets two or three weeks prior to the flight, they were always able to get us bassinet seats over the phone.

I suspect they would displace travelers without infants to different seats pretty much at any time, up to and including an hour before departure. They're very nice to parents and families.

To be honest, it would be doing those people a favor anyway, since the bassinet seats are the least comfortable on the plane. They only become great when compared with the alternative of a baby on your lap for 12 hours.