While providing players with as much information as possible is usually ideal, there are situations when it can hurt a game. One such case is when the players have perfect information -- that is, they know everything there is to know.
A good example is the game checkers, where the entire board and all pieces are visible. There are no elements of the game itself which are hidden from either player.
Nearly every game needs some element of surprise, and "pure" strategy games are the best example. In many cases, this element is provided by other players, be they human or AI. If you always knew exactly what your opponent's next move was, there wouldn't be a whole lot of tension!
Solitary games that lack an unpredictable opponent need some other way of spicing things up, and some form of randomization is virtually always the answer. The solitaire card game that nearly everyone is familiar with uses a randomized deck. If the card order was the same every time, there would be almost zero replayability.
However, it's not just the games without human players that are seriously damaged by perfect information. The aforementioned checkers was recently "solved" -- meaning if neither player makes a mistake, the end result will always be a draw.
Having some form of hidden information is a crucial element to preventing a single strategy from dominating. In many strategy games, there is a "fog of war" which covers the map, and exploration is necessary to reveal what lies in the darkness.
Sometimes even the map itself changes over time. For example, in the recent Civilization games, technological research reveals new resources. Their sudden appearance can greatly alter the "perfect" strategy for a given situation. This constant need to adapt to previously unknown circumstances is a big part of what makes games fun.
Hidden information is especially important in single-player games where AI opponents are simply executing lines of code written by a human programmer. In most games it is nearly impossible to develop an AI that will compete with the best of the best. If the human player is also able to see the entire game situation clearly, it's only a matter of time before the AI's patterns are learned, dissected and ruthlessly exploited. A game solved in this manner quickly loses whatever charm and joy it once held.
The amount of information that "should" be hidden can vary greatly, and ultimately depends on the preferences and goals of the designer. The card game Dominion makes 10 random action cards available to players at the beginning of each play session, and the inability to predict the order in which cards are drawn provides a great deal of replayability.
However, those who have played enough games will begin recognizing the optimum strategies for a given set of action cards, and Dominion has become formulaic for some. Adding more randomization or hidden information would probably improve the game for these players, but it might also make it less enjoyable for others. Hey, game design is more art than science.