Odds of Getting In
By around your junior year of college in this or a related major, you'll know whether or not you have what it takes. This is a prestigious field, and there may be some games you have to play, such as securing letters of recommendation from the top robotics engineering professors on campus. Then again, if you're smart enough to do good work in professional robotics, the recommendations should come easily.
Like an art or English student, you'll have to build up a portfolio that proves you're working on cutting-edge stuff that actually works. You should also cultivate a specific area of expertise that applies to a specific position—this will make you way more hire-able. Everyone needs a few experts on hand for when the going gets tough.
All of those objectives can be achieved by being proactive while still in college. There are lots of robotics companies that hire interns, so be on the lookout for opportunities to get some official working experience in the robotics world.