According to the announcement by Jarrod, the logic for the Sportsmanship badge only counts votes that a user casts after posting his own answer. I questioned that logic in a comment under the announcement; here's the gist:
It's not uncommon for me to scroll down a question, reading the answers, upvoting as I go... this badge logic might encourage people to answer before reading the existing answers, which is bad.
that's a good point about the Sportsmanship logic possibly discouraging reading all the existing answers.. we'll discuss it. (source)
Has this discussion happened? Just now, I went to a question, started writing an answer, upvoted the existing answer, remembered the timing rule, immediately removed my upvote, finished/posted my answer and then found that I couldn't re-upvote the first answer because it'd been 11 minutes since the initial vote was cast.
In short, I would like to officially feature-request this change to the Sportsmanship logic: allow votes you cast before you submit your own answer to count towards the badge.