An even harder problem than $t>2$ and $n=m$ is the Prouhet–Tarry–Escott problem. Now I leave it to you and google to find lots of examples ;-)
Wikipedia has a page on this, for example, but it has dashes in the link and I can't figure out how to link to it. Someone more mark-up savvy feel free to edit this.
Prouhet–Tarry–Escott_problem at Wikipedia.