The EE crew is correct, because server config ends up being an infinite number of possibilities - so anything they (or even I, as a layman) might be true…or not…for a certain config.
What I meant by my statement about configs….is that they MUST be set somewhat correctly….that is, the apache/php/mysql balance…..
From afar and from what your server admin said, the first settings to look at are the Mysql.
It is easy to see if mysql is being stretched just by running a free program or two….
http://blog.mysqltuner.com/download/
https://launchpad.net/mysql-tuning-primer/+download
Your admin can easily run these and get some results - they look at the past mysql logs and let you know if anything is way off.
An example….of me running one of those…
——————-
OPEN FILES LIMIT
Current open_files_limit = 1134 files
The open_files_limit should typically be set to at least 2x-3x
that of table_cache if you have heavy MyISAM usage.
Your open_files_limit value seems to be fine
TABLE CACHE
Current table_cache value = 512 tables
You have a total of 210 tables
You have 317 open tables.
The table_cache value seems to be fine
TEMP TABLES
Current max_heap_table_size = 16 M
Current tmp_table_size = 32 M
Of 1389207 temp tables, 2% were created on disk
Effective in-memory tmp_table_size is limited to max_heap_table_size.
Created disk tmp tables ratio seems fine
TABLE SCANS
Current read_buffer_size = 2 M
Current table scan ratio = 3166 : 1
read_buffer_size seems to be fine
——————————————————-
That would seem to be your fist step.
Chances are that PHP is fine…...but if Mysql does not show any problems, then you want to move on to apache (the thingy that serves web pages).....etc.
This has settings which tell it how many connections to handle at one time, etc. etc….
It all depends on the version as to what applies.
I think you should do the mysql checking…...before anything! Why complicate matters if most of the problem is there….
Of course, we assume you regularly optimize your tables…....