As is the answer to most everything where humans are involved, it depends.
If you are talking on line, I got my degree in computer science and we know that computer randomizers are good but not perfect. That said, in the course of weeks or months of daily online play, you would never be able to pick out the faults in the randomization process without some fairly sophisticated programming that recorded every revealed card during play.
In live play, I've read a study that said 7 good shuffles were needed to randomly redistribute the cards after a hand. If the cards go into an auto shuffler, that will randomize the cards. But what happens when the auto shuffler is broken or not installed at the table? The dealer will reuse the same deck, shuffle 3 times, cut the cards two or three times in the process. That is not sufficient. If he is a sloppy shuffler, which happens rarely, I'll ask him to shuffle again. If it is a home or underground game, the quality of the dealer and shuffle may be sorely lacking and who knows how bad the shuffle will be.
Does that insufficient shuffling result in anything that a player can take advantage of? I don't think so. If you knew every hand that was discarded the previous hand, you might occasionally be able to spot some sort of grouping on the flop. But I doubt it.
If the hand went to showdown, and if 4 players showed paint, and if it was on a painted board, and if the dealer was sloppy or lazy such that those painted cards tended to stay together on the shuffle, and if some of those same cards came out on the flop, you might not be surprised if the turn and river was some of those same painted cards. That's a lot of ifs.