The second you pass the minimum check in time on CO or any airline for that matter, you are at the mercy and discretion of the ticket agent, or if you decide to escalate the situation, the manager on duty or station manager.

I've only been told too late to check in once (I tend to show up early). In that case, I offered that I had only one carry on bag, there no line at security, and I was ready to run and was checked in.

Also, I've read threads in the past on various airline forums that basically indicate that you are more likely to be refused check in when the flight is full and they want to minimize the risk of an oversell situation.