Check-in rules

We arrived at JFK 51 minutes prior to flight departure and they denied us from boarding because I did not make the checked baggage cut-off of an hour.

First, is there a rule that says you can't check your bags in for a later flight?

Secondly, once you check in on-line 24 hours earlier, can they release your seat with 50 minutes prior to departure? I wanted to store my bags and just take a carry-on and stay on the same flight.

Any assistance would be appreciated.

Thanks!