In this work we provide a counterexample for Schur's Theorem on triangular matrices on infinite dimensional spaces. Moreover, the counterexample provided is a compact quasinilpotent operator. Indeed, the result neither depends on the index of the chosen basis for the matrix representations nor on the upper-lower choice for the triangular matrix. As a consequence, we see the optimality of a result by Halmos on matrix representations of operators. Namely, Halmos proved that each operator can be represented by a matrix with finite columns. Finally, we $\lq$answer' a philosophical question posed by J. B. Conway in \cite[p.213]{2}.