Yes - it's possible not to be able to win. In fact it's possible to not be able to survive until the second turn.
Here's an example of someone playing a game which they lost on the first turn: http://boardgamegeek.com/thread/373797/it-takes-talent-to-lose-on-the-first-turn
Maybe they could have done better, but if they had cards such that they couldn't treat anything in the black area then that would be the end of the game. Here's a full example:
Initial placement of cubes are three in Baghdad, Riyadh and Karachi, two in Cairo, Tehran and Delhi and one in Algiers, Istanbul and Moscow (i.e. 18 of the 24 black cubes are already on the board).
Position after initial infection
Four players, player 1 is Medic with Santiago and Beijing. The other players (Operations Expert, Scientist and Dispatcher) have no special events, and don't have Atlanta. The first player can't treat any cities in four actions, and picks up one city and one epidemic at the end of their go.
The epidemic city is Mumbai, the marker is moved up one space, and after shuffling the city cards Baghdad and Riyadh are drawn. These cause two large chain reactions with 18 outbreaks and requiring 45 cubes on the board - a loss by two methods.
Position after epidemic in Mumbai, and infection in Baghdad (outbreaks highlighted in green)
Position after infection in Riyadh
Pandemic can be a frustrating game at times. I think the author of the session review sums this up pretty well:
All that set-up, for less than a minute of game play!