Problem:

Sometimes near the last turn, I can't build a route, even if I'm sure I have the right number of cards.

Solution:

Make sure you have enough plastic trains left.

At the beginning of the game you are dealt 45. Each time you build a route, those plastic trains are laid out on the map. When you don't have those trains anymore, you can't build a new route. The number of trains you have is displayed on the bottom right corner of the screen along with your score.