If $(x_n) \in \ell^\infty$. According to Lorenz the Banch limit is unique (also known as *almost convergent*) iff
$$\lim_{p\mapsto\infty} \frac{ x_n + x_{n+1} + \cdots + x_{n+p}}{p} = L  \quad (*) $$
uniformly in $n$. Setting $n=0$ yields Cesaro summability.

As Aaron says, the converse is false. If each $x_n$ is chosen uniformly at random from $\{0,1\}$ then this sequence almost never has property $(*)$ (see Connor's appropriately named article *Almost none of the sequences of 0's and 1's are almost convergent*)

However the Cesaro limit of this random sequence $(x_n)$ is almost always $1/2$ by the law of large numbers.