You can set this per-user on a per-hour basis.

Under the WITH option for the MySQL GRANT command yu gave the following:

with_option:
    GRANT OPTION
  | MAX_QUERIES_PER_HOUR count
  | MAX_UPDATES_PER_HOUR count
  | MAX_CONNECTIONS_PER_HOUR count
  | MAX_USER_CONNECTIONS count

You can set

• queries per hour
• updates per hour
• inserts per hour
• concurrent connections (not per but at any given time)

This can give you finer per-hour granularity not only in terms of the number of connections made, but what can be executed.

The columns for these user grants are locate in mysql.user:

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mysql> select column_name from information_schema.columns
    -> where table_schema='mysql'
    -> and table_name='user'
    -> and column_name like 'max%'
    -> order by ordinal_position;
+----------------------+
| column_name          |
+----------------------+
| max_questions        |
| max_updates          |
| max_connections      |
| max_user_connections |
+----------------------+
4 rows in set (0.01 sec)

mysql>

I wrote about these things before

• May 24, 2011 : What are user connections - when are the created and destroyed?
• Jun 16, 2011 : Would it be sensible to create a MySQL user for each user account of a web application?
• Mar 22, 2012 : How can I limit MySQL connections?

Your Original Question

As shown above, you can set user per-hour user limits. However, I came to notice that you want just DB Connections open for any given user.

You may require some special customized authentication that counts how many connections a user has. In MySQL 5.1+ you could possibly script this:

SELECT COUNT(1) ConnectionCount,user
FROM information_schema.processlist
WHERE user <> 'system user'
GROUP BY user;

This would allow you to know the counts for any connected user.

You may want to look into connection pooling and persistent connections to regulate the number of users connected.