As soon as you have a sum of distinct $i$th powers, say $a_1^i+\dots+a_s^i$, equal to another sum of (not necessarily distinct) $i$th powers $b_1^i+\dots+b_s^i$ ($s$ is, of course, the same), you have the desired property for $n\ge\max\lbrace a_1,\dots,a_s,b_1,\dots,b_s\rbrace$. So, your question is about a "minimal" solution of $$a_1^i+\dots+a_s^i=b_1^i+\dots+b_s^i$$ in integers with $a_1,\dots,a_s$. The equation does not look pretty enough, and solutions for small $i$ can be found "by hand".
As soon as you have a sum of distinct $i$th powers, say $a_1^i+\dots+a_s^i$, equal to another sum of (not necessarily distinct) $i$th powers $b_1^i+\dots+b_s^i$ ($s$ is, of course, the same), you have the desired property for $n\ge\max\lbrace a_1,\dots,a_s,b_1,\dots,b_s\rbrace$. So, your question is about a "minimal" solution of $$a_1^i+\dots+a_s^i=b_1^i+\dots+b_s^i$$ in integers with $a_1,\dots,a_s$. The equation does not look pretty enough, and solutions for small $i$ can be found "by hand".