Tactical downvoting is not nice. However, just because you get a downvote as a new answer is added doesn't mean that the author of that answer is the one who downvoted, and further, a downvote from the author of a competing answer isn't always tactical.
To clarify: if I read an answer that I think is wrong, I will downvote it. In many cases I will then write an answer that I think is right. This is no coincidence. I'm not trying to suppress you or promote myself, it's just that I think you're wrong and I'm exercising a few of the things we do here on SE when someone is wrong on the internet.
Also, my answer will bump the question to the front page. This increases the chances that a whole pile of other people will suddenly read the question and its answers for the first time. If one of the answers is not good, one of these new readers may well downvote it.
Nobody likes downvotes. Your answer might not be the slightest bit wrong. But the most likely explanation for downvotes is that something is wrong with your answer, and having that vote correlate closely in time with a new answer doesn't change the probability of it being a "good" downvote rather than a tactical one.