No! Not only are some hands of Klondike Solitaire unwinnable, but in some deals, you cannot make even a single move.
TechUser.net explains in The Probability of Unplayable Solitaire (Klondike) Games:
The Solitaire game Klondike has a few idiosyncracies: not all Klondike games are solvable. Moreover, Klondike sometimes produces unplayable games. In such cases no moves are available to the player even at the beginning of the game. The probability of occurrence of unplayable games is an important number as it is a lower bound for the probability of occurrence of unsolvable games.
An unplayable Klondike game satisfies the following three conditions simultaneously:
- No aces are in the fifteen playable cards
- None of the seven playable cards in the row-stacks can be moved to a different row-stack.
- None of the eight playable cards in the deck can be moved to any of the seven row-stacks.
The article goes on to work out the math behind the probabilities, and explains that "anywhere from 2.5 to 10 percent of all Klondike games are unsolvable", and that "on average 0.25 percent (one out of 400) of all Klondike games are unplayable".
So now you know!