I think this can better be translated as 'despite' or 'in spite of' or 'even though' rather than 'although' or 'but' as there are many forms of 'but' which is rather confusing.
EG: Even though I dashed down the stairs/ despite dashing down the stairs, I missed the train by one minute.
If I`m not mistaken のに carries with it a sense that the result was contradictory to what one would usually expect. IE 「If someone is dashing down the stairs, of course he/she will be able to catch his/her train. yet dispite all that dashing he/she failed to catch his/her train」 そな通りですか
Erm. This is a JLPT3 entry isn't it? Not JLPT 2.
Yeah, ex #5928 does not make much sense as is.
#5928 The Japanese does not make sense. たまたま遅れなかったのに、means you were usually late. たまにしか遅れなかったのに、means I was rarely late. たまたま遅れたのに、means I happend to be late.
Ooops extra 'is' left in the see also after re-arranging the sentence.
I'm not that certain of what I typed now either - although the two are quite different usages.
i wonder if explanations should go in the see also or the comments. its a bit redundant having both, but I like to have the seeAlso be informative as I will have a list of just these at some point.
Before that ... I think a [delete] link for the see alsos would be nice ^^v