Can postgresql accept mutliple connections in the same instance?

I am running of postgresql database servers with generally 30-50 users
at a time per server. I have noticed one thing for web based databases
that they fail to initialse a pg_connection connection every now and
again and return no error message at all. Though one of hte developers
tells me that sometimes it gets a link resource allocation error on
opening connections to the database server which i dont know if that is
related. Max connections is set to 500, the webpage for the server code
initalises and then closes the connection when its done. Using netstat
on open postgresql connections seems to match with what apache is
showing as current connections.

This seems to happen sometimes when the backups are going, which are
done every hour though not always. Users notice this a lot between the
hours of 6am to 10am which 1 in 20 connections failing to be
established. Does pg_dumpall or pg_dump lock the database block access
to the database so you cant create connections? Could it be possible
postgresql can only create one connection at a time and if it gets two
requests for a connection at the same time it fails? Do you have any
ideas how I can test the database in a way to find out what could be
causing this problem?