If you go anaerobic - easily done with a punch bag or running uphill - you are going to run out of breath. That's normal. Try sprinting flat out on your bike and you'll be lucky to make it much past 15 seconds before your power output drops. Watch a Olympic 400 metre runner - 40+ seconds and they're done. Stopping recovering and going again is just doing intervals and intervals come in a variety of guises. If you want to do longer intervals, you go slower. Getting fitter won't allow you to do longer intervals, it'll just mean your power is higher and allow you to recover quicker. Fortunately intervals are a very good way of getting fitter - provided you have builty up good CV fitness through less intense efforts.