One database connection is equal to one web request (in case, of course, your client reads the database on each request). By using a connection pool these connections are pre-created, but they are still used one-per-request.
Now to some numbers - if you google for "Tomcat concurrent connections" or "Apache concurrent connections", you'll see that they support without any problem 16000-20000 concurrent connections.
On the other hand, the MySQL administrator best practices say that the maximum number of concurrent database connections is 4096.
On a quick search, I could not find any information about PostgreSQL.
Q1: is there a software limit to concurrent connections in PostgreSQL, and is the one of MySQL indeed 4096
Q2. Do I miss something, or MySQL (or any db imposing a max concurrent connections limit) will appear as a bottleneck, provided the hardware allows a large number of concurrent connections?
Update: Q3 how exactly a higher connection count is negative to performance?