In 1984 Boshernitzan proved an upper bound on the number of ergodic measures for a minimal subshift of linear block growth and asked if it could be lowered without further assumptions on the shift. In this talk I will discuss recent work (joint with B. Kra) in which we show that Boshernitzan's bound is sharp. Our methods allow us to study the, a priori, larger set of nonatomic generic measures, and one of our main results gives a bound on the size of this set for a minimal Interval Exchange Transformation.