Results of attempts to check baggage in early vary by airport and often by chance. MIA is not checking in bags more than 5 hours before a flight.

What one person was able to do at OMA does not reflect the possible situation for another passenger. OMA is a small airport but now and then I run into folks there who stick to whatever the rules of the day happen to be.

Rechecking in at LAX in the area just past immigration and customs (recheck area) has worked for me a few times even when there was a several hours wait for my next flight.

As far as rules of thumb go, there aren't any.