Your not "treating" it like 1600 iso film except from the actual taking of the picture. Pushing the film increasing film speed yes, but the most important part is increasing development time. If you take a roll of 400 Tmax, and push it to 1600, then develop normally you will get nothing but black negatives, because the film itself wasn't exposed long enough to light at normal development times to show up on the neg. However if you over develop, say 20% longer, then the developer has more time to eat away the unexposed silver and creates a high contrast version of the picture you took. The more you push, the longer you have to develop, and the more detail you lose.
Many people choose to push film rather then carry different speeds with them, and they decide to live with the quality loss associated with it.
Comsumer color films are not very pushable usually, Kodak films can usually be pushed only one stop, i.e. 200 to 400iso, and so on. Pro B&W films can be pushed up to 6400iso sometimes Tri-x 400 for example is very pushable, and i think looks better when pushed then when exposed normally.
Just remember though, B&W pro film require special development that Walmart photo shops and the like usually don't carry. Sometimes they are stupid and try to dev. them in color chems, that makes for an interesting roll. If you don't do it yourself, you have to take it to a pro shop that can dev. them for you, and make sure you tell them exactly what you did to the film so they can compensate for your push.