Yes cornmeal does go bad. It contains a little oil in it that can go rancid over time. If refrigerated, it can be stored for about 4 months. The best way to tell if it is rancid is to smell it, if it is bad, it will smell stale.
Cornmeal will eventually expire, it does have a considerable shelf life but doesn't taste very good if it is stale. If you have had it more than a year, even if it doesn't smell or look bad, throw it away.