We do have this problem! It is only a matter of how small the PGBouncer machine is compared to the PostgreSQL. In our case the PGBouncer had to deal with 16000 req/s, coming from 700 clients going through 100 connections to the PostgreSQL.
We now have 4 PGBouncers for this DB, and we're 'load balancing' them with DNS.
I'm guessing it's a situation where you are running a large volume of very easy queries, for example primary key lookups with no joins?