All i can advise to you is as follows.. at the end of the season or maybe now, do an honest review of yourself. Do you find yourself weaving in and out of traffic, do you obey traffic rules, do you speed exessively? Just be truthfull with yourself and you will find the answer. Only you can determine if a higher performance bike is safe for you.
I think this sums it up very nicely...its exactly what I did. Matter of fact, I didn't even bother selling my 250 until the beginning of the following season (in case I really did end up deciding on keeping it).
Mind you, I'm not about to say I have the greatest amount of experience on a 250 either (about 5000km within 2-3months of riding) but I definitely felt extremely comfortable with it. Not just riding it in a straight line either but turning as well as braking (hard).
I evaluated myself multiple times over the winter season like sleezyray mentioned and ended up grabbing a slightly modded ZX6R. Coming from the 250, first thing I noticed; it was a hell of a lot easier to go into corners a lot faster than you should and like someone else mentioned in this thread you'll definitely be a lot slower going into those curves/corners before you start taking it on slightly faster (or braking a lot closer to the turn). I wouldn't say it took a lot of time for myself, but from my own experience I believe it just comes down to how often you ride to gain that experience. Secondly, I noticed I had to be more quicker at checking my surroundings compared to being on the 250. It didn't take long to pick up but it definitely felt different from the previous bike. And finally, I spent time focussing on braking hard, accelerating/deccelerating (and understanding how 'twitchy' it could feel), figuring out my ideal RPM range I wanted to cruise around the streets with, and same applied to the highway (FYI felt a lot better being on the highway on a 600 compared to a 250). This coupled with hours & hours of riding have made this season very enjoyable (although work has been cutting into my riding time this summer
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My own experience on 600 SS bikes have been on my own ZX6R as well as a 600RR (amazing bike btw) so I can't speak too much in regards to the R6. My only piece of advice is its extremely easy to become overconfident on your 250 with time and experience. Whatever you do, don't bring that overconfidence to whatever bike you upgrade to next.
With that said, if I were to do it all over again - I would have personally started with a STOCK SS bike. But no regrets.
Sorry for the long post, but I hope this helps with your decision as to what you could expect with a 600 SS (coming from a 250). Good luck & ride safe!