Cameron et al. determined the maximum size of a null subsemigroup of the full transformation semigroup T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} on a finite set X and provided a description of the null semigroups that achieve that size. In this paper we extend the results on null semigroups (which are commutative) to commutative nilpotent semigroups. Using a mixture of algebraic and combinatorial techniques, we show that, when X is finite, the maximum order of a commutative nilpotent subsemigroup of T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} is equal to the maximum order of a null subsemigroup of T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} and we prove that the largest commutative nilpotent subsemigroups of T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} are the null semigroups previously characterized by Cameron et al.