Probably useful: $3^3+4^3+5^3=6^3$.

Let me use @straight's idea to show a construction—only missing point is finding an example for $n=4$. Let $1\le x_1<\cdots<x_n$ be such that \[ x_1^3 + \cdots +x_n^3 = y_n^3. \]Using the fact $3^3+4^3+5^3=6^3$, we thus obtain \[ (3x_1)^3 +\cdots+(3x_n)^3 +(4y_n)^3+(5y_n)^3 = (6y_n)^3, \]giving an example of $n+2$. As we already have $n=3$ taken care of, it suffices to find an example for $n=4$.

@above example for $n=4$: $6^3+5^3+1^3+1^3 = 7^3$.

@straight you need the example to be of distinct integers, I think the smallest possible one is $2^3 + 3^3 + 8^3 + 13^3 = 14^3$. For $n=3$ one can also take $1^3 + 6^3 + 8^3 = 9^3$.