In the documentation there is a list of possible values that textStatus might be if the error() callback fires ("timeout", "error", "notmodified" and "parsererror") but nothing specified for the success() or complete() events.
Are there any other values besides "success" that could be passed to the success() callback? Is this documented anywhere?
From looking at the source it appears that notmodified could be a value, although I've never seen it.
I don't see documentation per se, but you can find it in the source. Here's a nifty viewer.
http://james.padolsey.com/jquery/#v=1.4&fn=jQuery.ajax
Other possible values are "parsererror" and "notmodified".
Success and complete are documented at here. You don't really need to worry about what textStatus is passed to the functions, it's all handled automatically. The success function is called when/if the call succeeds, and complete is called at the very end of everything else.
So, as an example from something I'm working on, I chained together three functions that are triggered when you click a section of the page: a div fades out, and when that's done it runs an ajax call. The success function updates the content of the div (while it's faded out), and then the complete function calls fadeIn on the div. So what it does is it fades out, updates, and fades back in with new content.
Success can only be called on success. So the easy answer is no.
Complete will always run, regardless of what how ajax did. So complete should be able to get and values textstatus can return.
On the forum I found textstatus can be any of these values
"timeout"
"error"
"notmodified"
"success"
"parsererror"
http://forum.jquery.com/topic/jquery-ajax-textstatus-documented