For the family of model soft-core Coulomb potentials represented by V(r)=-[Z/(r(q)+beta(q))(1/q)], with the parameters Z > 0, beta>0, q >= 1, it is shown analytically that the potentials and eigenvalues, E-nu l , are monotonic in each parameter. The potential envelope method is applied to obtain approximate analytic estimates in terms of the known exact spectra for pure power potentials. For the case q= 1, the asymptotic iteration method is used to find exact analytic results for the eigenvalues E-nu l and corresponding wave functions, expressed in terms of Z and beta. A proof is presented establishing the general concavity of the scaled electron density near the nucleus resulting from the truncated potentials for all q. Based on an analysis of extensive numerical calculations, it is conjectured that the crossing between the pair of states [(nu,l),(nu',l')] is given by the conditions nu' >=(nu+1) and l' >= (l +3). The significance of these results for the interaction of intense laser field with an atom is pointed out. Differences in the observed level-crossing effects between the soft-core potentials and the hydrogen atom confined inside an impenetrable sphere are discussed.