We give a criterion for checking the Cohen-Macaulayness of the tangent cone of a monomial curve by using the Gröbner basis. For a family of monomial curves, we give the full description of the defining ideal of the curve and its tangent cone at the origin. By using this family of curves and their extended versions to higher dimensions, we prove that the minimal number of generators of a Cohen-Macaulay tangent cone of a monomial curve in an affine l-space can be arbitrarily large for l ≥ 4 contrary to the l = 3 case shown by Robbiano and Valla. We also determine the Hubert series of the associated graded ring of this family of curves and their extended versions.