Yes, if you use up all your upload speed on your stream and don't leave enough left over for your game, it'll affect your ping. Remember your audio bitrate is there too, so if your audio is set to 128, your video is set to 1800, and your ISP upload rate is 2000, you'd only have 72 left over for your game. Which obviously wouldn't be enough.
I'd set it up experimentally, trying different bitrates until you find the highest bitrate under your provider's (Twitch?) cap that has no affect on my gameplay. When I had 5000 up I set it to 3000, for example, because any more than that and I would occasionally feel some lag (my ISP may have been unreliable about giving me the full 5000).