I've taught myself to play several instruments over the course of my life and I at least can play at some more.  Every one of them has helped with all the others to at least some degree. 

I feel the big payoff is if you do any jamming, improvising or writing.  There is a sort of cross-pollination of ideas between instruments, and anything you learn on one may give you new ideas of things to do on the others.  The basic elements of music like intonation, ear, timing, dynamics and theory, I think every instrument a person can play helps to develop at least some of those to some degree.  Since those really are basic elements, they help a bit with everything you play on every instrument you can play.  It adds up.

Besides, ideally, you should always end a practice or play session at a good moment.  If you stop when things are sounding poor and you are frustrated, it makes the learning process harder.  One of the tricks to practising instruments is to learn to bring your practice/playing to a close at a good strong moment you can be proud of or feel really good about.  That's what makes a student eager for the next practice, and that is what helps you keep up your drive and momentum.  If you maybe miss that precise moment on violin (easy to do, especially at first), if you have another instrument you play, you can pick it up and kick some rump for a little while so that the session ends with a good strong positive moment.  

If you don't do that, and you just beat your head into the instrument to where you can't do anything right?  You are practising suckiness.  You don't want to do that.  LOL 

To clarify just a little, if you stop when you feel frustrated and beat down, you take that feeling with you from the practise and you don't feel good about your playing.  Probably it lasts for at least some hours.  You'll think of yourself as someone who can't play for crap.  That can build over time into kind of dreading the time to practise and play.  That is how some instruments end up going cheap on ebay or collecting dust in a closet. 

But if you can manage to always bring practise or playing to a close at a good moment, where you can think "Yes!  I RULE!  I love playing this instrument!" or at least "It is coming right along.  Getting better every day." then your time with your instrument(s) is one of the best times of the day.  You're likely to take the extra moment to check tuning, wipe down your instrument and put it away thinking "Later, baby.." and anticipating the next time you'll be able to pick it up. You'll spend hours thinking of yourself as someone who can play.  You won't miss many practices even on busy days, if you can possibly find some time.

Maybe just my point of view on practise and playing, but think on it.  I've only been playing violin for a little over a year, but I've played other instruments to over 35 yrs.  I've noticed a few things in that time that at least seem to work for me.  Your mileage may vary, of course.