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. It was shown by Connor (in his appropriately named article) Almost none of the sequences of 0's and 1's are almost convergent that randomly generated sequence of 0's and 1's almost never has property $(*)$ ; but almost every randomly generated sequence of 0's and 1's is Cesaro summable to 1/2 by the law of large numbers.