From personal experience, and now talks with our ND, it matters more how those children are treated and the expectations of them and especially how you word things.
My personal experience was this - I was a bright child, tested years above my peers in reading, writing, etc. My parents sent me to catholic school bc the public schools in our area were starting to get 'rough' and the first of the magnet schools wasn't yet built. Let's just say it wasn't in the budget to allow some students to advance while other got more help, so I was a very bored little girl, and thankfully shy enough that I didn't turn my boredom into destruction and managed to stay out of trouble. But one too many years of being told how 'smart' I was and how I 'didn't need to pay attention like the other kids', and when I was finally out of that environment it hit me hard. The first time I encountered something difficult for me - freshman algebra - I was at a loss. I'd never been pushed before. Never had to deal with anything being difficult. And I completely crashed under the pressure. Fast forward to college, yes I made it through, and even managed a few A's here and there in those classes that came 'easy to me'...but for everything else I struggled. To the point I was on academic probation for a semester. So from the smart kid who was allowed to sit in the office sorting papers or hanging out with the nurse just because I didn't feel like being in class and graduated middle school as salutatorian, to the nervous reclusive wreck stuggling just for a passing grade (D is for diploma became my mantra at one point), I know all too well how easy it is to go wrong and let that smart kid down.
Our DD is 13mos and far brighter than I was at her age. She can speak about 50 words - clearly and accurately - understands probably 100 or so more. Her ability to reason and speak in phrases is on par with most of our friends' 2 and 3yr olds. I don't want for her to end up in my situation and therefore am planing to send her to a montessori school that will allow her to progess at her own pace. But more importantly, in talking with our ND, she suggested we not place emphasis on telling DD how smart she is, but to focus on and praise her efforts and how hard she tries. And as a dog trainer that's not something I often have the chance to do (well, except for maybe the owners of the dogs I work with) - my focus is always on the end result, not so much how we got there. So I'm learning how to encourage her in different ways that are a bit foreign to me, but will hopefully pay off in the long run. To keep her from someday becoming overwhelmed, like the first time she approaches a task that doesn't come easy.