In 2004, Kim and Chan (KC) reported a decrease in the period of torsional oscillators (TO) containing samples of solid (4)He, as the temperature was lowered below 0.2 K [Kim E, Chan MHW (2004) Science 305(5692):1941-1944]. These unexpected results constituted the first experimental evidence that the long-predicted supersolid state of solid (4)He may exist in nature. The KC results were quickly confirmed in a number of other laboratories and created great excitement in the low-temperature condensed-matter community. Since that time, however, it has become clear that the period shifts seen in the early experiments can in large part be explained by an increase in the shear modulus of the (4)He solid identified by Day and Beamish [Day J, Beamish J (2007) Nature 450(7171):853-856]. Using multiple-frequency torsional oscillators, we can separate frequency-dependent period shifts arising from changes in the elastic properties of the solid (4)He from possible supersolid signals, which are expected to be independent of frequency. We find in our measurements that as the temperature is lowered below 0.2 K, a clear frequency-dependent contribution to the period shift arising from changes in the (4)He elastic properties is always present. For all of the cells reported in this paper, however, there is always an additional small frequency-independent contribution to the total period shift, such as would be expected in the case of a transition to a supersolid state.