I'm a veteran here, I guess. I have four gifted kids, all of whom started violin at fairly young ages.
Eldest dd started at 3.5 and is HG/PG.
Middle dd and ds started a year or so later and are HG.
Youngest dd started at not-quite-3 and is probably HG/PG.
All started within the Suzuki paradigm. All took between 4 and 7 months to be ready to move off the first piece (which is really six pieces: a theme and set of five variations). Two of them started on pretend violins. It didn't make me run for the hills. If you haven't experienced it, it can sound daunting, but it isn't months of working to perfect a piece that they can only play rather badly at the outset. It's a case of starting out with the most basic of (silent) skills that are layered one on top of another as each solidifies. I didn't feel frustrated: I could see my kids progressing, the layers of skills being accured atop each other, and for the most part they enjoyed the process so the journey was what mattered, not the goal. The first few lessons were about postural and fine-motor skills. After two or three weeks they started working on getting a nice sound on open strings. After two or three months they were ready to start stringing pitches together, playing in-tune notes with the left hand fingers, crossing strings, getting a staccato sound here, a legato sound here. Each accomplishment is exciting. I didn't spend my time waiting for them to be competent enough to play that first piece on a recital: I was too busy noticing all the hundreds and hundreds of steps along the way. Getting myself free of a goal-oriented mentality was essential, and freeing.
On the other hand, i think Geofizz is right: the learning is not terribly developmentally appropriate for very young children. Playing the violin involves an exceedingly complex skill-set. Playing that first piece ... well, 90% of the basic violin skills have to be well-learned to play that very first piece well. The one advantage I see in starting children on violin before the more traditional age of 6 is that they are much more process oriented and don't have the impatience of older kids. They tend to be willing to set up and consolidate the early skills properly before moving on. If their parents aren't communicating impatience to them, anyway.
My eldest took the longest to move off the Twinkles (7-8 months). She is stubborn, has very high needs for autonomy, and was relatively goal-oriented even at 3, and she wasn't willing to try much until she was sure she had learned it internally and was ready to do it well. That went for all the tiny steps in her early violin learning too. There was some longer-term fall-out, I think, from the tension between her desires and the her developmental limitations for the complexity of integrating all those intellectual, musical and fine-motor tasks. Even though she was incredibly bright and incredibly focused, it took her a long time to find her stride on the violin in terms of owning the process, motivating her own learning, knowing why and how she was best to accomplish the day-to-day stuff. Really she was 12 before she hit consistently smooth sailing. She's now playing professionally and part-way through a performance degree and a highly sought after college music conservatory program. So it all came out in the wash, but it was a challenge, one that might have been avoided to an extent if we had waited a little longer before starting.
My middle two did much better through those early years. I think partly because of the later start. And also I was more laid back about making it work. I knew I had to be fully committed to supporting them, but I had less anxiety about expressing that support in exactly the right way. I became really convinced (still am, actually) that around age 4.5 is the best age to start lessons on the instrument, even for intellectually gifted kids.
But alas my youngest child would have no part of that plan. I completely agree with Geofizz that there's nothing wrong with telling a child that they're too young for something, but my poor 2.75-year-old had to come and watch her three older siblings have violin lessons every week, and observe their immensely fun multi-level group classes, and we *had* a hand-me-down tiny violin in our house already just waiting for her, and she knew what practising every day looked like and felt like. There came a point where it just seemed cruel to keep telling her that she couldn't join in on what everyone else was doing. And so eventually the violin teacher and I both caved in and allowed her to start her own little 10-minute lessons. And because she had in essence had years of introductory teaching by observation she actually moved fairly quickly through the early steps. She was off a pretend violin within 3 or 4 lessons, and playing all the Twinkle variations within 5 months.
Still, she was a special case: she'd already had years of being steeped in the pre-learning of her Suzuki beginner skills, and it was impossible to keep her waiting in a family full of children doing exactly what she wanted to be doing. Furthermore, as she's grown up it has become apparent that she is this incredibly unique kid who has not only a very high level of intellectual giftedness but an unbelievably cheerful, resilient and creative attitude to challenge -- and none of the emotional intensity that had come to seem normal to me as part of the gifted-kid psyche.
I do want to make a pitch for Suzuki teaching. If you find a good Suzuki teacher you will not find formulaic snail's-pace teaching; you will find someone with a framework and skill-set that allows for excellent highly individualized creative teaching within the developmental limitations of children younger than 6. For some children some skills will come very slowly; for others they'll come quickly. A good Suzuki teacher will take great joy in the accomplishments of both groups of kids, and will not be attached to particular expectations for rate of progress. And you should also find within the Suzuki program a multi-age, multi-level community of fellow learners and similarly supportive parents.
One thing I also wanted to mention is that traditional teaching tends to rely on gradual mastery of music-reading skills from the beginning, and despite all signs to the contrary, intellectually precocious kids may not be ready for note-reading as young as is expected of them. My eldest was reading proper novels at 4, but note-reading did not really click for her in a meaningful way until she was 8 and into student concerto repertoire on both violin and piano. My ds also had a considerable lag in his note-reading skills. Youngest dd began reading music fairly well around her 5th birthday, but she was already heading into intermediate repertoire at that point. Reading on-instrument is considerably more abstract and complex than reading language, because there's no one-to-one correspondence with anything: that blob means one thing if it's there, something different if it's there, something different again if there's a sharp way over here, and has a different value entirely depending on the numbers at the beginning of the piece and depending on how many beams are above it and the whether you're counting in half-beats or beats, etc. etc.. The Suzuki approach allows kids not to be held hostage by their developmentally appropriate lags in note-reading ability: the approach allows them to continue to be challenged musically even when their decoding isn't yet up to the task.
Having said all those nice things about the Suzuki approach, it's not a rigid training system that qualifies and certifies the best teachers. There are crappy teachers who consider themselves Suzuki teachers, some of whom have loads of "training" and experience, there are people who are wonderful teachers who believe many of the same things philosophically and pedagogically as good Suzuki teachers but without calling themselves Suzuki teachers. And there are great teachers of all stripes who probably won't jive with your child's personality. So the most important thing is to find the teacher who is right for your child and whatever age and stage you're at.
Keep an open mind. Consider waiting a year. Don't lie if that's the route you take, just say "The best teachers we can find don't really know how to teach children as young as you are. They want you to be older, so we will plan on starting when you are 4." In the meantime, observe lessons, recitals, group classes, fill your home with opportunities for general (non-violin) musical experimentation and exposure.