If driving with multiple filters, your vehicle will lose performance. Your car will have a greater restriction in the intake system from having to pass through two separate elements. You would, however, have cleaner air coming into the system, but at a cost of the performance. You'll have to have more gas pedal to compensate which would cause other issues as has been explained.
Aftermarket manufacturers do have what is called a "cold air intake", which can replace the entire system from the air box and filter all the way to the throttle body. This would be used as a single filtration system (only one filter). Depending on the manufacturer, this type of filtration system can actually perform better in both flow and in how well it cleans the air (you'll have to do your homework to figure out which are which). This can provide you with a small performance boost (it's actually freeing up power which was is already there, but that's another story), as well as better protect your engine from particulate matter. The mass air flow (MAF) sensor, manifold absolute pressure (MAP) sensor, and oxygen (O2) sensors will tell the computer how to adjust for these variations so as to give the engine the fuel it needs to stay at its target air/fuel ratio for optimal efficiency (given the fuel map in the computer).
Some will tell you that the manufacturer will put together the optimum system and fuel mapping for your vehicle. I would tell you this is a complete fabrication. Unless you are talking about high performance vehicles, vehicle manufacturers use parts and settings which are a trade off between comfort, economy, and performance. They will give you parts which will work well for your vehicle, but are cheap to mass produce, and provide you the most comfort from your vehicle (whether it is noise or whatever). There is always room for improvement over what comes stock. Fuel mapping is done conservatively on the rich side so as to prevent dreaded engine knocking or pinging. There's even room for improvement in high performance vehicles. The conservative nature of the fuel mapping is also done so you get better longevity out of your engine. I will say fuel mapping has improved over the past decade so as to provide better fuel economy, but as I said, there is still room for improvement over what comes stock in almost any situation.