A class of states of the electromagnetic field involving superpositions of all the excited states above a specified low energy eigenstate of the electromagnetic field is introduced. These states and the photon-added coherent states are shown to be the limiting cases of a generalized photon-added coherent state. This new class of states is nonclassical, non-Gaussian and has equal uncertainties in the field quadratures. For suitable choices of parameters, these uncertainties are very close to those of the coherent states. Nevertheless, these states exhibit sub-Poissonian photon number distribution, which is a nonclassical feature. Under suitable approximations, these states become the generalized Bernoulli states of the field. Nonclassicality of these states is quantified using their entanglement potential.