  Let $E$ be an elliptic curve over $\mathbb{Q}$ described by $y^2= x^3+ Kx+ L$
where $K, L \in \mathbb{Q}$. A set of rational points $(x_i,y_i) \in
E(\mathbb{Q})$ for $i=1, 2, \cdots, k$, is said to be a sequence of consecutive
cubes on $E$ if the $x-$coordinates of the points $x_i$'s for $i=1, 2, \cdots$
form consecutive cubes. In this note, we show the existence of an infinite
family of elliptic curves containing a length-$5$-term sequence of consecutive
cubes. Morever, these five rational points in $E (\mathbb{Q})$ are linearly
independent and the rank $r$ of $E(\mathbb{Q})$ is at least $5$.
