My experience is that they suddenly become a lot more common when I'm trying to fish for something specific, like perch. :D
In my game, this seems to happen a lot too. When I get a quest to catch certain fish (i.e. Sunfish, Red Snapper), then I seemingly get buckets-full of all the other fish available in the area while that specific fish that I need for the quest seem to be as rare as legendary fish. Even if I get quests for certain Gems (Ruby, Topaz), those specific gems seem rare to encounter in fishing treasure chests and gem nodes from any form of mines. Oddly enough, I haven't had that same type of "rarity effect" to deal with on Clint's quests to mine a certain amount of a certain type of ore.
I'm beginning to think that there is a hard-coded "artificial rarity spike" that takes effect when you get quests for a specific item.