Yes! It happens to a small percentage of students every year. In fact, in the last few years, the number of residency slots total throughout the country was less than the number of medical school grads (including FMG's). Of course, the competitiveness of the residency may influence whether or not you go unmatched. AMA/AAMC and AOA/AACOMAS should be able to provide recent statistics.
If you don't match, you either wait a year, or in some instances you can apply to programs that have vacancies open. Not sure, however, the way that one does that.
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"Be wary when one says impossible, for it most likely means that they have just never seen it or done it before..."
