Generally speaking, you will start to feel out of breath when you exceed your AT. At that point your aerobic metabolism cannot keep up with your energy expenditure and your body will try to get more oxygen, but that's not possible, so you feel out of breath. Your HR will probably also increase rapidly to try to get more oxygen to your muscles -- also fruitlessly.
So if you are exercising to the point of being out of breath
and you have ME/CFS, you are probably doing yourself harm. ME/CFS patients get PEM from exceeding their AT (again, generally speaking).
If you exercise to the point of being out of breath, but don't get PEM (the one with "prominent symptoms in the neuroimmune regions" and exacerbation of flu-like symptoms),
and you can increase your activity level over time by doing this, it is unlikely you have ME/CFS. You probably do have exercise intolerance, which is no small thing, and should have that investigated by your physician. Some common causes of exercise intolerance are cardio-respiratory conditions, various forms of orthostatic intolerance, and deconditioning.
Very, very few (if any) ME/CFS patients can run for 25 minutes without getting out of breath. Many of us can't even walk for 5 minutes without going over our AT. As an example, at my last CPET test I was at 96% of my AT just sitting on the bike before the test even started.
Needless to say, it didn't take much very light pedaling to get to my AT. The thought of running is just laughable.