You do not have to run the NICS check if your state has its own instacheck system or if your concealed handgun license meets the ATF requirements to supercede the NICS check.
So basically, it is a state-by-state thing. Some states meet the requirements and some don't. As an example, the ATF recently notified Georgia that because it had not kept up with recordkeeping requirements, their CHL was no longer valid as a substitute for NICS.