A few times a year. I wouldn't be surprised if it follows a similar pattern as kids who go to daycare: more illnesses than kids not in daycare, but fewer when reaching school age years. Maybe more in the beginning of residency and less at the end (entirely speculative and not evidence based)?
I typically get a relatively mild cold 1-2 times per year and get really sick only every couple of years (Once during my intern year with possibly flu, once in fellowship, and once earlier this year when I was sick for almost three weeks). For better or worse, I was able to work through most of those illnesses except for 1-2 days total of illness in the latter two.