  We study kth order systems of two rational difference equations
$$x_n=\frac{\alpha+\sum^{k}_{i=1}\beta_{i}x_{n-i} +
\sum^{k}_{i=1}\gamma_{i}y_{n-i}}{A+\sum^{k}_{j=1}B_{j}x_{n-j} +
\sum^{k}_{j=1}C_{j}y_{n-j}},\quad n\in\mathbb{N},$$
$$y_n=\frac{p+\sum^{k}_{i=1}\delta_{i}x_{n-i} +
\sum^{k}_{i=1}\epsilon_{i}y_{n-i}}{q+\sum^{k}_{j=1}D_{j}x_{n-j} +
\sum^{k}_{j=1}E_{j}y_{n-j}},\quad n\in\mathbb{N}.$$ In particular we assume
non-negative parameters and non-negative initial conditions. We develop several
approaches which allow us to extend well known boundedness results on the kth
order rational difference equation to the setting of systems in certain cases.
