Yes, some questions are too anecdote-y, but I don't think that anecdotes directly correlate to bad questions. Given the nature of this site, and the desire to answer "practical, answerable questions based on actual problems that [users] face", the questions almost must include anecdotes to set the stage for the real question.
However, plenty of work is still left to be done to ensure that users are structuring questions in such a way that the anecdote supports the question rather than buries it. By "plenty of work" I mean "show good models" -- either by suggesting edits, editing the question directly, or otherwise talking about the issue here.
I try to do this when I see a good question buried in a too-localized anecdote -- I edit it to highlight the question and call out the supporting anecdote via formatting, or sometimes the edits are more significant. But I don't do it consistently (I'd never get any work done, if I did!), unfortunately, and that's not helpful because it doesn't show consistent models.
It is easy to edit out extraneous details, but I don't think we can draw any sort of hard line about what is too much, because like a lot of things at The Workplace (for good or for bad), it depends.
I don't think we're headed down the same road as early ProgSE, for no other reason than all three of us -- and many of our users -- know what that road was and want to avoid it.