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Upgrading to a better violin isn't going to make you sound instantly better unless your own playing ability is already close to getting the best out of what you already have. It will have better tone, which you may notice even when it's being played by a near-beginner, and it will respond better to subtle changes in technique. I don't know anyone who is under any illusion that a Strad would make them a much better player.
That said, an upgrade can make it much easier to learn once you get past the basics, because your ears are so important to learning the difference between good and bad technique. A better violin will reward incrementally better technique with a more obvious improvement in sound. This is why you can hit a ceiling on a beginner-level violin as a beginner, but a professional violinist can pick up the same beginner-level violin and sound far better. The professional violinist has technique fine-tuned for years on an instrument that rewards good technique more, and can take that technique back to a lesser instrument. By contrast, the beginner has to search for improvement on an instrument that isn't as responsive to changes in technique, and has less-trained ears to begin with -- and if you have a hard time hearing what's better, it's harder to practice good technique. That's why, once you have the basics down, it is usually worthwhile to have the best (sometimes most expensive, but often not) violin you can get within your budget.
And, of course, the pros will still hit a ceiling on that beginner-level violin. It's a much higher ceiling, so they'll sound pretty good on it, but not nearly as good as they sound on the much more expensive instruments they normally play.
This is why people on the pro audition circuit talk about a minimum price level for a professional violin. It's because they're all so good that it really does matter. When there are a hundred people auditioning for one spot in an orchestra (this is not an exaggeration, it's actually a low-end number for even part-time pro orchestras) and every single one of them has the ability to get the best out of a violin in that price range, that's the point where you need an instrument at least comparable to what others are playing. Of course, most of us are not aiming for that. It's a completely different world for people who aim to play for a living.
In my experience, at a certain level (I'd say upper intermediate or higher), changes in equipment can make big differences. Last fall, I switched from Hill Dark to Jade rosin. It wasn't really a planned switch; I dropped my rosin cake on a concrete floor and needed new rosin urgently, and my local shop didn't have Hill Dark in stock. But the difference was instant. My agility and articulation on the lower strings improved immediately. I can still get good results with inferior equipment. I still keep my first student bow around as a backup bow and sound decent with it. A couple years ago I accidentally took the wrong bow out of my case for a concert and played with my cheap student bow until intermission, and it went fine. But there is a very noticeable difference when I use my main bow.
I appreciate all of the great thoughts here on this. I'm probably saying something similar. My path was and still is one of trial and error. What works and what doesn't work as well.
I played for awhile and noticed, wow I feel like I am pulling teeth to get a strong sound from my violin. Seems I need to fight the bow to get sound. Either that or I'm going deaf. I tried different bows. different violins, listened to other players and tried to objectively see how they were doing it with much less effort. I heard one experienced player in particular say that he basically just sits the bow on the strings when he plays. He doesn't use much pressure at all. I looked at myself and realized I was using a lot of pressure and it seemed to be killing my bow technique. Nothing worse than the feeling that I was fighting to get what I needed from the instrument.
Then I realized that yes indeed, the equipment can make a difference. Some of it was and still is me. I don't own any particularly nice violins or bows, at least when looking at what I paid for them. I've tried many different combinations , different strings, moving the sound post and bridge around, different rosins.
I feel I'm getting it narrowed down to what I like over time. I know part of it is me and part of it is I should probably get something better eventually. Slowly I'm beginning to find a place I can be content with. In the interim I have found ways to work around some of it.
My ears sometimes seem fickle. I was trying to record myself last night and tried two different violins and several bows. Different mic positions. None of it sounded good. I packed up and went to bed. Maybe today will be different.