This study aims to explore the possibility of detecting indices that could potentially provide warning about the proximity of internal damage to critical levels, beyond which catastrophic fracture is impending. In this direction, advantage was taken of the Cumulative Counts that were recorded during the mechanical loading of specimens made of either plain or fiber-reinforced concrete. The parameter adopted for the analysis was the average rate of change in the Cumulative Counts. Τhe evolution of the specific parameter was considered in the Natural Time Domain, rather than in the conventional time domain. Experimental data from already published three-point bending protocols were used. It was revealed that the specific parameter attains, systematically, a limiting value equal to unity exactly at the instant at which the load reaches its maximum value, which is not identical to the load recorded at the instant of fracture. Similar observations were made for a complementary protocol with uniaxially compressed mortar specimens. The conclusions drawn were supported by the b-values analysis of the respective acoustic data, again in terms of Natural Time. It is, thus, indicated that the evolution of the average rate of change in the Cumulative Counts in the Natural Time Domain provides an index about the proximity of the applied load to a value beyond which the specimen enters into the critical state of impending fracture.