For a recent student seminar at my university, I and my colleagues were asked to provide three "tools" that every professional musician requires in order to succeed. It's rather difficult to narrow things down to three items, but here is my set, in no particular order.
(1) Technical versatility. No matter your instrument, to be a successful professional in the 21st century you must be ready and able to serve in whatever capacity you are needed. If you are a singer, you need to move easily among opera, choral, church, and recital settings. A clarinetist has to be able to blend with her section, tune to the strings, make confident musical decisions in a recital, and be flexible enough to adapt to different conductors. A pianist needs to know how to adjust to playing with singers, choral ensembles, orchestras, and instrumental soloists, all of which have differences in onset of sound, types of conducting, balance/voicing requirements, etc. These things require you to have a comprehensive command of your instrument and the ability to adjust your sound production, tempo, inflection, etc. at a moment's notice. (You also have to know how to remain healthy and balanced in your life, but that's a topic for another day.)
(2) Mental resilience. There are several different components here, but the main point I'm getting at is the ability to recover quickly from emotional highs and lows, remaining focused and able to continue working. Music is powerfully intertwined with emotions. However, there is a danger of becoming overconfident after a great experience and sinking into despair after a bad experience (performance, audition, lesson, or even just an interpersonal interaction). You have to develop the proverbial "thick skin," able to glean useful feedback from each experience while discarding self-doubt, self-praise, attempts at manipulation or power plays by others, or similar distractions. Obviously, this is INCREDIBLY difficult - it's actually a lifelong process of learning by experience (there's that word again). I'm certainly not perfect in this area by any means, but I have seen enough to know how important this skill is. Those who have this "grit" (see: Angela Duckworth's book) will be able to accomplish some remarkable things, and those who lack it often crash and burn - even if they have considerable natural talent.
(3) Communication skills. I can't possibly emphasize this one enough. You need to communicate things clearly, concisely, and comprehensively in both verbal and written mediums. That means no spelling errors, no grammatical errors, no loose ends to conversational threads, no unanswered questions or vague wording, etc. You may have phenomenal ability and technical mastery, but if you can't interact smoothly and graciously with other people you will greatly reduce your chances of holding a job, getting asked to work with an ensemble again, recruiting and retaining students, or helping your organization build its donor base. This also applies to working with people from other cultures. If you're learning their language, learn it correctly, thoroughly, and humbly. Know their social cues as much as possible so that you don't come across as rude or condescending. (Side note: this also applies within the US! Midwesterners and Southerners are really quite different from coastal folks!!) Music is an amazingly diverse profession, and you never know which personal connection might result in that elusive "next big break"!
Finally, and somewhat paradoxically since I've been preaching a bit, is intellectual and musical humility. No matter how high you rise, you've got to remember that you're not all that, and that being a musician doesn't somehow elevate you above "common people". Keep an eternal perspective, especially after very positive or negative experiences. Always look to learn from musical interactions, and be willing to adapt your interpretations or preferences (this is especially important for those working with conductors!!) for the sake of collaborative harmony. Be joyful and thankful for as many things as you can find - sunrises, working gas stations, great coffee, and the privilege of working in music. A spirit of grateful humility, combined with rejoicing in beauty, is really what underlies all of the above "keys" to succeeding in music.