Abhyankar defined an ideal to be Hilbertian if its Hilbert polynomial coincides with its Hilbert function for all nonnegative integers. In 1984, he proved that the ideal of (r+1)-order minors of a generic p x q matrix is Hilbertian. We give a different proof and a generalization to the Schubert determinantal ideals introduced by Fulton in 1992. Our proof reduces to a simple upper bound for the Castelnuovo-Mumford regularity of these ideals. We further indicate the pervasiveness of the Hilbertian property in Schubert geometry.