Is it possible for someone to run a mile in 4 minutes?

Yes, these are the Outdoor and Indoor Records by both men an women.

= 1 Mile Middle Distance Record = Men's Outdoor

3:43.13 - Hicham El Guerrouj from Morocco in Rome, Italy on 7/07/99

Men's Indoor

3:48.45 - Hicham El Guerrouj from Morocco in Gent, Belgium on 2/12/97

Women's Outdoor

4:12.56 - Svetlana Masterkova from Russia in Zürich, Switzerland on 8/14/96

Women's Indoor

4:17.14 - Doina Melinte from Romania in East Rutherford, New Jersey on 2/09/90

* http://www.trackandfield.com/records/world/1-mile-middle-distance-record/

Yes, but you would have to be very fit. Or have a very active imagination