Odds of Getting In
Finding a paleontology job will take more than just blind luck. Each year there are bunches of freshly degreed paleontologists trying to claw their way into some sort of paid employment.
The key is to make yourself more marketable so you can move yourself to the top of the rock pile. Internships or work/study gigs show employers you're not afraid of real-world challenges—in fact, some employers might even require one of those more hands-on jobs before they'll hire you.
You'll probably find more job openings in industry-based research and university settings than in museums that fund research work. Don't overlook teaching at the high school or junior college level either, as those jobs offer the predictability of classroom work with few research requirements.