I am primarily a developer but I also like to understand the big picture, including software design and UX but also system administration, infrastructure, hardware architectures, and everything else that *directly affects the software I develop*.
Deep understanding of the big picture is key to developing quality software, IMO. You need to understand what comes ahead of you (requirements, business needs, etc) and where your work is headed afterwards... The best way to understand it is to wear these hats from time to time or have previous work experience in those fields. When recommending candidate for developer positions, someone who has system administration experience is a bonus.
Yes, many days I need to take on multiple hats and switch gears as shit comes up in prod and I need to fix a config on production servers or assist whoever has the hands but lack the knowledge. That's the start-up culture I guess, even though I work for an established 100+ year old company.