I don't think this is correct. At least in my sport (cross country/track and field), I've known multiple people who have run for one undergrad and a different grad school. Two things: 1) You only get 4 years, so these people didn't participate a full 4 years during undergrad (redshirt, disinterest, etc) 2) Once you start using your eligibility, you have 5 years to use it all. This can change if you get a medical redshirt approved by some board at the NCAA.
I've known of a 4:07-4:10 miler who graduated undergrad with one year left. He went to law school across the country, and almost broke 4:00...I guess those lawyers don't need to study much