You can beat airlines. Mistake fares and fares sold below cost definitely exist, though they're a lot less common than they used to be as pricing models have improved. You're more likely to see them if you pay attention to off-season and new routes that aren't popular. Severe weather predictions and similar events can also create large price drops. I once got a $20 flight to Hawaii by simply buying just before a typhoon that didn't hit.
A friend of mine recently got a too-good-to-be-true fare from Europe to Asia on Qatar, and the ticket was promptly cancelled and refunded by the airline when they found the error.
