In the hope of avoiding model dependence of the cosmological observables, phenomenological parametrizations of Cosmic Inflation have recently been proposed. Typically, they are expressed in terms of two parameters associated with an expansion of the inflationary quantities matching the belief that inflation is characterized by two numbers only, the tensor-to-scalar ratio and the scalar spectral index. We give different arguments and examples showing that these new approaches are either not generic or insufficient to make predictions at the accuracy level needed by the cosmological data. We conclude that disconnecting inflation from high energy physics and gravity might not be the most promising way to learn about the physics of the early Universe.