In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential non-local model and another deals with the strain gradient theories. The present paper has the objective of establishing the fact that the length scales present in non-local elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nano-scale. The numerical results based on the new non-local strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical non-local stress model and the contemporary strain gradient theory. Thus, this higher-order non-local strain gradient model provides an explanation to some observations in the classical and non-local stress theories as well as: the strain gradient theory in these aspects.