In addition, it will very much depend on the location.
In Sydney for example, I doubt that the DL counter opens much more than 3 hours before the SYD-LAX flight, as there are no other DL flights. On the other hand, in ATL, I'm pretty sure that you will find a place to drop your bags from early (5 am or earlier) until close to the last flight of the day.
Cheers,
GenevaFlyer