First off get the 600 out of your mind, or you won't be living very long. You don't have the skills for a supersport race bike, which is what a newer 600 is. You have to learn to walk before you can run, so stick with the 250.
Now onto your main concern, the short answer is YES, however, that is for an experienced rider, you being inexperienced then your answer is NO! You will have more than enough problems trying to learn to ride in good weather, and believe me, you will have plenty of problems with that, there is MUCH to learn about riding, and most of it comes with experience. Then as you become an experienced rider, you can start to get into other bad situations, like riding when it is cold and riding in the rain, which is NOT fun at all.
So, for you being a beginner, take an MSF course first thing, then get the good gear, then pick up a used 250cc bike and practice. Forget about the 600 and riding in bad weather, you won't survive, or you'll scare yourself so badly you'll give up riding, not to mention would have to change your underwear after a serious scare, if you survive.
I used a Ninja 900ZX as my sole transportation for awhile, I lived in So. Cal. right by the beach, so the weather was not a huge problem. But even then when it was cold, it was BAD! I would lay on the tank tucked down behind the windscreen and tried to get as much of the engine heat as I could, wasn't much help. In the rain things really get dicey, any painted lines are SLICK, as are manhole covers, the absolute worst though are the car/truck drivers, in the rain some don't slow down, and a cyclist is at the mercy of the car drivers at that point. On dry pavement at least an experienced rider has a chance and can control the traffic pattern much better, in the rain you lose that advantage due to speed/acceration constraints.