When cucumbers are left on the vine they will get big, heavy, bitter, and possibly even woody, making them quite inedible. If you’re looking to keep a cucumber to store seeds for the next year, you can let one grow until it starts to dry out. But you need to make sure you pollinate the flower yourself and then close it off, or you may end up with some kind of hybrid as cucumbers can cross pollinate with squash and some melons.