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 ;-)

[http://en.wikipedia.org/wiki/Prouhet-Tarry-Escott_problem](http://en.wikipedia.org/wiki/Prouhet-Tarry-Escott_problem)