If f is a binary word and d a positive integer, then the generalized Fibonacci cube Q_d(f) is the graph obtained from the d-cube Q_d by removing all the vertices that contain f as a factor, while the generalized Lucas cube Q_d(lucas(f)) is the graph obtained from Q_d by removing all the vertices that have a circulation containing f as a factor. The Fibonacci cube Γ_d and the Lucas cube Λ_d are the graphs Q_d(11) and Q_d(lucas(11)), respectively. It is proved that the connectivity and the edge-connectivity of Γ_d as well as of Λ_d are equal to ⌊ d+2 / 3⌋. Connected generalized Lucas cubes are characterized and generalized Fibonacci cubes are proved to be 2-connected. It is asked whether the connectivity equals minimum degree also for all generalized Fibonacci/Lucas cubes. It was checked by computer that the answer is positive for all f and all d≤9.