The goal of the present work is to obtain a reliable estimate of the masses of the ground and radially excited states of fully-heavy tetraquark systems. In order to do this, we use a nonrelativistic model of tetraquarks which are assumed to be compact and consist of diquark-antidiquark pairs. This nonrelativistic model is composed of Hulthen potential, a linear confining potential and spin-spin interaction. We computed ground, first, and second radially excited cc (c) over bar(c) over bar and bb (b) over bar(b) over bar tetraquarkmasses. It was found that predicted masses of ground states of cc (c) over bar(c) over bar and bb (b) over bar(b) over bar tetraquarks are significantly higher than the thresholds of the fall-apart decays to the lowest allowed two-meson states. These states should be broad and are thus difficult to observe experimentally. First radially excited states are considerably lower than their corresponding (2S-2S) two-meson thresholds. We hope that our study may be helpful to the experimental search for ground and excited cc (c) over bar(c) over bar and bb (b) over bar(b) over bar tetraquark states.