  Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0 = 0, F_1 = 1$
and $F_{n+2} = F_{n+1}+F_n$ for $n \geq 0$. In this paper, we solve all powers
of two which are sums of four Fibonacci numbers with a few exceptions that we
characterize.
