It's difficult to say how many messages the backend can handle and this depends a lot on the component and the type of traffic. The backend is optimized for handling "usual" LoRaWAN traffic, which means that the Router will be able to handle thousands of messages per second, and efficiently filter the ones that are meant for TTN and forward them to the corresponding Broker(s). In the Broker there are a number of things that influence the capacity and performance. Next to the message rate, other factors are the number of registered devices, the number of devices that use the same address and the number of gateways that receive a transmission. The performance of the Handler will depend mostly on the message rate.
Until now we have been focusing on building functionality for the network, and not on handling thousands of messages per second. Our complete staging environment is running in a number of docker containers on one small, single-core VM. It is able to handle usual traffic just fine, but won't survive an attack or load test.
We are currently working on the next iteration of the backend, which will include a number of things that should already improve the performance of the backend and make some big steps towards better scalability and distribution of the network.
The end goal of TTN is that the backend is distributed over multiple TTN operators (organizations that run a part of the backend, such as Routers and Brokers) and regions (like datacenter regions).
Some network components will be able to scale horizontally, so that an operator can instantiate a number of them behind a load balancer and scale out when network load increases.