If you've put T-Max 400 in the camera and set the ASA/ISO dial to 1600, when the camera's meter indicates '0' at 1/250 @ f/2.8 you're already underexposing by two stops. This is because the meter thinks you've got 1600 speed film in the camera, but the film in the camera is still 400 speed. It doesn't magically change its chemical properties just because you changed the ASA/ISO dial.
That's why you need to "push" the film in development: Your film will already be two stops underexposed if you follow the meter's recommendation when the ASA/ISO dial is set two stops faster than your film really is.
If you change the shutter time from 1/250 to 1/125, the meter will show one stop of overexposure (because you've told it there's 1600 speed film in the camera) when the exposure value of 1/125 @ f/2.8 will still be one stop underexposed for 400 speed film in those lighting conditions.
If you also open up the aperture to f/2, you've increased exposure another stop and the meter, set for 1600, will show two stops overexposed at 1/125 @ f/2. Your film will be exposed "properly", based on the fact the meter thinks your film is two stops faster than it really is. If you set the ASA/ISO dial to 400, under the same lighting conditions the meter would say '0'.
No matter what the ASA/ISO is set to and what the meter says, exposing at 1/125 @ f/2.8 in the same lighting conditions will always result in the same actual amount of light hitting the film.
Think of it this way: The ASA/ISO dial does nothing to the film in your camera. What it does is change the amount of light your meter will read as '0'. The faster your meter is told the film is, based on the position of the ASA/ISO dial, the less total light, determined by the combination of aperture and shutter time selected, it will tell you the correct exposure is.
If you are then going to compensate your shutter time and/or aperture to make up those two stops, you won't need to "push" development because your film will not be underexposed. You're then using the same shutter time and aperture that you would have used if the ASA/ISO dial was set to 400 and you lined up the meter at '0'. In which case it's a whole lot easier to just set the dial to 400 and follow the meter's recommendation.