In the book "The Hunger Games", it is mentioned many times that the boy and girl from the same district, like Katniss and Peeta from District 12, are enemies, because they eventually have to kill one another. But we see people forming teams - so why can't a boy and a girl work as teammates for some time? It looks like a natural strategy.
Tl;dr - There have been, with one exception ever, no official alliances or teams in the Hunger Games. Nonetheless, contestants are free to form temporary alliances and frequently do so.
So yes, two contestants can definitely work as teammates for a time. They are also still technically enemies and will suffer no external game penalty for betraying their ally at any time.
Now, as to why most contestants don't do this... it's because most of them are inexperienced and untrained. An alliance only helps if your ally is useful, and most of the contestants are pretty much useless. So a savvy contestant will know that they still shouldn't team up with their district buddy. And an unsophisticated contestant won't think of the option and figure out how to try and actually make it work.
The high-powered districts have lessons on how to form alliances, how to break alliances, and other such things, so they can act pretty smoothly within that framework and their contestants are prone to creating giant deathball alliances to change the odds from 1/24 to 1/8 or better.
While Jertnarg explained it in terms of "usefulness", I'll stick with trust. Or lack thereof.
It is very, very difficult to trust someone to guard your back when, in order to save her own life, she will need to kill you at some time. In order to make that work, you need an extraordinary faith in your ability to read your partner's intentions, and to kill him just before he decides to kill you.
As a watered-down example of the process, you should watch the first season of "Survivor".