I was playing cards (a variant of Gin Rummy) with some people recently. When it was my turn to shuffle, I would try to do a very good job of it. I wanted the cards to be well-randomized.
We all noticed that when I shuffled, it took longer for someone to win the round. We assumed that the more random the cards, the harder it would be to get triplets or straights. You see, after a round, when we collected the cards, they were bunched together (people had laid down triplets and straights, the cards in their hands were collected in partial groups). Before the shuffle, they were not random. So a quick shuffle meant, obviously, that there was less randomizing, the bunches tended to stay together just a bit more for the next round. A card in play was more likely to have a "partner" card in play nearby. And more likely to be only 2 or 3 cards away, which is what was needed for the same person to get both cards on the deal.
OK, there's nothing radical about this analysis. But what was interesting for me was that one of the players started asking me to shuffle less. She won more with fewer shuffles. My guess is that there are two overall strategies, play as if the deck is random and play as if it is not. She had learned (maybe subconsciously) how to play with a strategy of non-random cards and this was successful whenever she played. However, that strategy did not work with more random cards.