  For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is
defined by the initial values $0,0,\ldots,0,1$ ($k$ terms) and such that each
term afterwards is the sum of the $k$ preceding terms. In this paper, we search
for powers of two of the form $F_n^{(k)}+F_m^{(k)}$. This work is related to a
recent result by Bravo and Luca concerning the case $k=2$.
