In the paper, we present sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables for a sum of entire functions, where $\mathbf{L}:\mathbb{C}^n\to\mathbb{R}^n_+$ is a continuous function, $\mathbb{R}_+=(0,+\infty)$. They are applicable to a very wide class of entire functions because for every entire function $F$ in $\mathbb{C}^n$ with bounded multiplicities of zero points there exists a positive continuous function $\mathbf{L}$ such that $F$ has bounded $\mathbf{L}$-index in joint variables. Our propositions are generalizations of Pugh's result obtained for entire functions of one variable of bounded index.