Can I claim a route longer than the number of trains I have left?
No, you cannot claim a route longer than the number of plastic trains you have left.

Ticket to Ride Rules


  Claim a Route
  The player may claim a route on the board by playing a set of Train Car cards 
  that match the color and length of the route and 
  then placing one of his colored trains on each space of this route...  
  
  Claiming Routes
  ... When a route is claimed, the player places one of his plastic trains in each of the spaces of the route...