History seems to show that well made violins can sound better, over time, as the wood ages - assuming proper care. Certainly, well made violins can get worse over time if not stored/maintained properly.
The chance of a poorly made violin (that presumably sounds like it was made) blooming and reaching the level of a well made violin violin over time is, well, negligible.
If I don't have time for a short post, I'll write a long post - (adapted from Mark Twain)
If I remember Julian Bream's autobiography correctly, a guitar spends 5 years improving, then 5 years playing well, then it slowly goes downhill. But that's just a thin, flat sheet of spruce, and also it sounds so anecdotal that it should be taken with a pinch of salt, and also we can assume it's a guitar that gets played at least 4 hours every day. I read somewhere that a lot of Strads are played out now and some people are preferring Amatis because they've been hammered less. But that's after 300 years.