Noah Nygeny currently holds the world record for the 1000m at 2:11.66. The 1000m is a seldom run distance, but will occasionally be run at a meet with special exceptions.
I think they can, but it would require some planning. While both has shown once in a generation mile talent, neither have shown top speed in the 800m. Both runners rely on strength to beat their opponents, and the 1000m needs an extra gear.