Is having too many multiple simultaneous AJAX connections bad?

IE6 isn't going to be your only problem; other browsers also limit the number of concurrent requests to the same server. Here's a good roundup, which says that as of that writing the defaults were:

Browser           HTTP/1.1    HTTP/1.0-------           --------    --------IE 6,7            2           4IE 8              6           6Firefox 2         2           8Firefox 3         6           6Safari 3,4        4           4Chrome 1,2        6           ?Chrome 3          4           4Opera 9.63        4           4Opera 10.00alpha  4           4

Aside from that, two other important quotes from that article:

It’s possible to reconfigure your browser to use different limits.

and

Note that IE8 automatically drops back to 2 connections per server for users on dialup connections.

...and for all you know, other modern browsers do, or may start doing so with their next "dot" release.

If you can, definitely try to keep the number of long-standing open connections to a minimum. Certainly don't actively keep more than one connection open for a long time.

If you're just doing a lot of individual, quick connections and sometimes they bunch up, you probably want to serialize them yourself rather than relying on the browser to do it. Have a queue of objects representing the request that needs to be done, and code responsible for doing them (one at a time).

too many is always bad.

Read more about AJAX optimization in this article.

Depending on the browsers listed in above answers, yes. In some cases if they are not queued to send a request in a chronological order you might find in certain scenarios where some returning information might get mixed up and show up in the wrong element looking to load an ajax POST request return information.