Take the suggestion even less than a grain of sugar

Actually it's a good idea to look what's in that file (reading it should be possible for ordinary users), and what "ulimit -n" prints. This should be possible from a CGI script, without ssh access.

If the number is something like 1024, we can be almost sure that there is a programming mistake in the OP's scripts. However, if the provider chose to lower that number to something like 10 (to prevent excessive use of resources), the web software might need some restructuring to make sure that file handles (including sockets) are never open longer than necessary.