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While discovering things is great fun, having no understanding of what you're finding is no fun at all. There was no clear pattern to the particle "zoo" that was being populated, and so no idea why nature would allow such things that were unstable and disappeared. That's not to say they disappeared without a trace, but rather the heavier particles always disintegrated into some set of lighter particles, all the way down to the very lightest stable particles, such as the electron.
After many years of experimental discovery, theory finally seemed to catch up with some potential answers in the 1960's. More particles meant more data, and more data meant that sooner or later a pattern was likely to be found, and that is exactly what happened. Just as the chemical elements were discovered and categorized by property, leading to the periodic table and the notion of atomic substructure, the new particles could also be categorized too - but only when certain new properties were theorized to exist, which were based on speculative concepts. At the heart of the new theory was, as might be expected at this point, further substructure.
In the atomic case, it was the patterns of the periodic table, and the phenomena of charge and electricity, which suggested some kind of atomic substructure, and the observation of the electron then confirmed it. In the case of the new entities created in accelerators, despite patterns that suggested further substructure, there was no sign of a telltale particle that did the same confirmation job as the electron had done decades earlier. Many of the new particles certainly seemed to be made of smaller pieces based on the patterns, but nothing that could be a single one of those pieces was ever seen on its own.