I do not really know for questions, but I believe one can delete ones questions alone if it is open and there is not too much activity on it (one or two answers with a few upvotes will result in prohibiting deletion); for closed ones, I think owner can only cast a vote to delete (like 10k+), after some time which takes effect if there are three.
However, if under some of your non-accepted answers you cannot see a 'delete' next to 'edit' and/or pressing it has no effect, this definitely seems like a browser issue. (I never used Chrome.)
]]>Later: Having searched more I see mention of "the delete button which shows up when I mouse over my answer" At least in Chrome I don't get that.
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