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

From the [rules][1], 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.


  [1]: https://cdn.svc.asmodee.net/production-daysofwonder/uploads/2024/05/7201-T2R-Rules-EN-20240423_LOWRES.pdf