No, you are limited to the 45 trains you started with.

From the rules, under Claiming a Route (emphasis mine):

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

If you do not have the trains available to place on each space needed then you may not claim the route.

This is also how Days of Wonder's electronic implementation works.

No, you are limited to the 45 trains you started with.

From the rules, under Claiming a Route (emphasis mine):

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

If you do not have the trains available to place on each space needed then you may not claim the route.

This is also how Days of Wonder's electronic implementation works.