As more nitrogen (or any other inert gas) is added to a flame, the flame temperature drops and the oxidation reactions cannot proceed fast enough to keep going and sustain the flame. this point is known as the:

Lean Flammability limit ,or lean limit.

Explanation:

The point is know as Lean Flammability limit ,or lean limit. The lean limit is usually expressed in volume percent. It can be defined as the lower range of concentration of over which a flammable mixture of gas and vapor can be fired at a constant temperature and pressure.

Here in this case also As more nitrogen (or any other inert gas) is added to a flame, the oxidation reaction stops as the concentration has dropped below the  Lean Flammability limit.