We discuss nonlinear dynamic models for the fluctuational opening of the base pairs in DNA and show that a standard model which is satisfactory for time-independent properties has to be improved to properly describe the time scales of the fluctuations. The existence of an energy barrier for the closing of the base pairs has to be taken into account. This introduces a model which sustains a new class of Intrinsically Localized Modes (ILMs). We investigate their properties numerically, and then consider two simplified versions of the improved DNA model allowing an analytical study of some properties of those ILMs. The models are different because the effective barrier necessary for the existence of this new class of ILMs is obtained either through the on-site potential or through the nonlinear stacking interaction, but they nevertheless sustain similar nonlinear localized excitations. An extension of the usual anti-continuum limit has to be introduced for the analysis, and relies on a continuation of localized equilibria from infinity.