I logged in to reply to this actually.
I played badminton and swim, my sister played hockey and we BOTH play instruments. I have a clarinet and saxophone (the trombone was on loan - free - from band). My sister has a trumpet and a french horn and loans a tenor horn from band (she used to loan a woodwind instrument as well).
We've traveled the country for camps and competitions (marching competitions as well as music). My sister went to China.
My husband plays guitar and drums (owns a drum set and several guitars), one of his friends plays guitar (also owns several).
As someone else said above, I have excellent lung capacity, stamina and posture, because of band. Not to mention attention span, focus and self control.
The benefits of music have been well established. Just type "benefits of learning music" into a search engine and you will see.
That's not to say sport isn't good either, but you know all the benefits of that so there's no point in spelling that out.
I would definitely encourage learning a musical instrument as well as playing sport. Some of my most favourite memories from school are my music lessons and "jamming" with my teacher and singing in choir (and winning competitions of course :D)