Particle filters algorithms approximate a sequence of distributions by a sequence of empirical measures generated by a population of simulated particles. In the context of Hidden Markov Models (HMM), they provide approximations of the distribution of optimal filters associated to these models. Given a set of observations, the asymptotic behaviour of particle filters, as the number of particles tends to infinity, has been studied: a central limit theorem holds with an asymptotic variance depending on the fixed set of observations. In this paper we establish, under general assumptions on the hidden Markov model, the tightness of the sequence of asymptotic variances when considered as functions of the random observations as the number of observations tends to infinity. We discuss our assumptions on examples and provide numerical simulations. The case of the Kalman filter is treated separately.