Is every game of solitaire winnable? The short answer is no.
It depends a great deal, however, on which game you are talking about, since there are many hundreds of different solitaire games. A lot of the time, though, people asking this question are asking it of the game that is commonly called Solitaire, but is really the game Klondike, the game that comes with Windows.
So a better question is:
Is Every Game of Microsoft Solitaire Winnable?
The answer is No. Many games are impossible and will block quickly. Estimates vary, but the consensus is that between 79% and 92% of games are winnable (meaning between 8% and 21% cannot be solved), depending on the exact rules by which you play.
In Klondike in Pretty Good Solitaire, setting the rules to 3 card deal and KingOnly (only Kings allowed in spaces) would be the same game as Microsoft Solitaire. Setting the rules to a 1 card deal and turning KingOnly off, however, results in very nearly every hand being winnable.
When referring to other games of solitaire, most people are interested in FreeCell. So another question is:
Is Every Game of FreeCell Winnable?
The answer is No. But in this case, it is close to yes. Unwinnable games in FreeCell are very rare. In fact, out of the first million games in Microsoft FreeCell (and also Pretty Good Solitaire), only the game numbers #11982, #146692, #186216, #455889, #495505, #512118, #517776, and #781948 are known to be unsolvable. Beyond the first million deals, less is known about which deals are unsolvable, but of course more unsolvable game numbers exist.
Beyond FreeCell, a few other solitaire games have been studied for winnability. For the most part, most games have unwinnable hands. Spider, for example, while the vast majority of games appear winnable, has a few unwinnable possible hands. Very few, if any, solitaire games are absolutely always winnable with every possible hand.
So, again, the general answer to the original question is no, not every solitaire game (or hand) is winnable.