I'm also a full-time programmer but I got my degree in music. I'm finding the best way to make it work is to be disciplined and schedule a specific amount of time. This is a skill I learned in school, as I was a composition major so I had to have a concentration instrument (which for me was the double bass), pass piano proficiency, pass sight-singing/aural skills, and be in a laboratory band every semester. The labs gave me a chance to also pick up better singing in chorus, learn to play a few ethnic drum classes (South Indian and West African) and I lived in a musician dorm so I also played guitar/bass in local bands. I had played brass instruments in high school so I helped out with local brass quartets/jazz bands as well.
GOOD NEWS/BAD NEWS:
Ultimately you need to schedule time to practice the instruments you'd like to learn, and be warned that the more instruments you know, the harder it is to stay "fresh" on a large number of them. Yes, it's like riding a bike, but for certain instruments (particularly brass/woodwinds/vocals) you'll find that your chops suffer every day you're away from the instrument, because the fine muscle groups required for those instruments are very specific in strength and agility. After a month away from trombone, for example, I found it very hard to hit the low "pedal" tones in the lower positions. A friend of mine is a saxophonist who plays in three ensembles for at least 40+ hours a week. None of them required him to practice altissimo (overblowing) techniques, and after only a WEEK of not using them, he completely lost a good chunk of his high range.
As I've learned from experience, "mastering" an instrument (as TomWij says) technically takes 10,000 hours or more (for more on this read Malcolm Gladwell's book "Outliers"). Most musicians who focus in performance stick to a schedule of 3 hours every day for 10 years in order to get this mastery. I don't think it's realistic to "master" more than one instrument every 10 years if you have a full-time job. But if you start practicing a lot of them, I think you'll find some skills do in fact cross over, and what the experts consider "mastery" may not be what you in fact want. So how you do this depends on you, but there are some tricks...
TIPS and TRICKS
As they say, to get to Carnegie Hall: practice, practice, practice. You have to be absolutely diligent. Keep in mind, however, that 30 minutes a day is completely reasonable for learning a new instrument. If you can devote this EVERY DAY without fail for at least a year, that instrument should be playable for your purposes. I do this sometimes, right before bed I'll realize I haven't played an instrument... so I'll crawl out of bed and set up something and play. It doesn't have to be long, but it teaches you the "craft" of musicianship.
Give yourself as many excuses as possible to play. Join a local band, a community group, a church, anywhere people will let you play with them. If you're in a city, find a corner where you can play. This will keep your skills sharp, and you can count this as time working on skills that cross over between instruments, like timing, intonation, and so on.
Speaking of which, try to surround yourself with musicians who you think are better than you. Yes, it sounds counter-intuitive, but if you can find people who are patient, they will bring you up to their level. This will be frustrating and most likely embarrassing, but it will only help you in the long run.
Invest in private lessons. You don't have to go every week, or even more than once or twice, but having someone check your form will ensure you're not missing some major issues that could slow you down or even cause long-term injuries from devoted practice.
Don't give up. Ultimately, it's going to be hard to stay inspired to do this as the years drag on. But you have to remember that it's a lifetime process. A bad day, week, or even year of learning can be overcome and even reversed in a very short time.
Break a leg, and stick with it!