  In this paper, we completely solve the Diophantine equations $F_{n_1} +
F_{n_2} = 2^{a_1} + 2^{a_2} + 2^{a_3}$ and $ F_{m_1} + F_{m_2} + F_{m_3}
=2^{t_1} + 2^{t_2} $, where $F_k$ denotes the $k$-th Fibonacci number. In
particular, we prove that $\max \{n_1, n_2, a_1, a_2, a_3 \}\leq 18$ and $\max
\{ m_1, m_2, m_3, t_1, t_2 \}\leq 16$.
