Many linux distributions ship with very low limits (e.g. 250) for the number of open files per process. You can use "ulimit -n" to see the current value on your system (be sure to issue this command in the same environment that your tornado server runs as). To raise the limit you can use the ulimit command or modify /etc/security/limits.conf (try setting it to 50000).
Tornado's HTTP server does not (as of version 3.2) close connections that a web browser has left open, so idle connections may accumulate over time. This is one reason why it is recommended to use a proxy like nginx or haproxy in front of a Tornado server; these servers are more hardened against this and other potential DoS issues.
