Subset sum-like problem over boolean vectors
Let us say that $x \leq y$ if $x_i \leq y_i$ for all $i$. Suppose that your starting vectors are $x_1,\ldots,x_n$ and your target is $y$. If $x_i \not\leq y$ then $x_i$ cannot belong to any solution, so you can throw away all these vectors. The remaining vectors OR together to a vector $x \leq y$. If $x \neq y$ then there is no solution. Otherwise, take all remaining vectors – this is clearly the maximal solution.