I'm going to keep this brief, but it comes down to technology and the creation of synthetic dyes. I can't speak for the 20th century, but I can get you through the 19th century.

My answer is coming mainly from two sources. The first is The Secret Lives of Color, by Kassia St. Clair. It's a really accessible book, with short profiles covering the history of dozens of colors. If this is a subject you're interested in, I can't recommend it highly enough.

The second source is Fashion in Paris: The Various Phases of Feminine Taste and Aesthetics from the Revolution to the End of the XIXth Century by Octave Uzanne. You can find both the original French and a 1901 English translation by William Heinemanm for free on Google Books.

Vibrant natural dyes, though not neons, have been highly prized throughout history. Tyrian purple, produced with snail mucus, was incredibly valuable in antiquity. In ancient Rome, its use was restricted to political and military leaders by "sumptuary laws."

Indigo is another great example of a vibrant natural dye that was valued across cultures and centuries. If you're interested in its history as a cash crop, there's a good article available on the Internet Archive-- "Eliza Lucas Pinckney: Indigo in the Atlantic World," by Eliza Layne Martin.

While they may not be true neons, I would point to the creation of aniline magenta (1859) and mauveine (1856) as important progenitors. I don't know much about the chemical processes responsible, but there was a boom in the discovery and mass manufacture of synthetic dyes in the late 1850s and early 1860s. Wearing dresses made with bright synthetically-dyed material became a status symbol, but a more accessible one. There were no sumptuary laws, and so these proto-neon clothes became desirable to members of the burgeoning middle class.

Octave Uzanne wrote about this period of French fashion from his vantage point at the turn of the 20th century, and he was less than impressed (p 129): "It would be difficult, indeed, to discover more violent contrasts of color, or shades more contrary to the laws of harmony, than those which enjoyed so great favor during the Second Empire. Even now, alas, occasional specimens of this hideous taste may be seen in shop windows. How can such overwhelming violets, such cruel pinks, such glaring greens... such blinding yellows, ever have left the dyer's hands? But all these cheap oleograph tints were received with acclamation, and reds of every kind, solferino, marengo, sang de boeuf, and so forth, enough to drive all the bulls in Andalusia into madness, were constantly invented, and greedily purchased."

Again, the usefulness of my answer depends on your definition of neon, and my knowledge of dye stops at the 20th century. Hopefully, however, this has at least answered your question a little bit. Let me know if you have any other questions!