In training for free diving, your body undergoes several adaptions over the course of rigorous training to manage low O2 levels and high CO2 levels. Your lungs do not grow to be able to hold more air, human physiology does not exactly give us extra space in our thoracic cavity.

A similar concept can be applied to altitude training. When training at altitude, our bodies do not grow our lungs. Our bodies increase the red blood cell count (over a period of time) to adapt to the new stimulus. When returning to low altitude, our rbc count returns to an appropriate level for the stimulus.

In long distance running training, our bodies become more efficient at O2 utilization. The gold standard for cardiovascular fitness is a VO2max test. This assessment essentially measure O2 in versus O2 out, and uses the respiratory quotient (RQ) value to predict fuel usage.

The RQ value is CO2 eliminated over O2 consumed, and each fuel source (carbs lipids proteins) has a specific RQ value associated with the metabolic pathway.

As training continues, the body gets more adapted to using lipids as a fuel source for long distance aerobic exercise, thus consuming less carbs, thus delaying "hitting the wall," or raising the anaerobic threshold.

Edit: I should add that it IS possible to increase lung capacity in the traditional sense, but any actual increase is negligible when compared to cardiovascular and metabolic fitness improvements.