Numerous sufficient conditions for adiabaticity of the evolution of a driven quantum system have been known for quite a long time. In contrast, necessary adiabatic conditions are scarce. A practicable necessary condition well-suited for many-body systems has been proven recently in [Phys. Rev. Lett. 119, 200401 (2017)]. Here we tailor this condition for estimating run times of quantum adiabatic algorithms. As an illustration, the condition is applied to the adiabatic algorithm for searching in an unstructured database (adiabatic Grover search algorithm). We find that thus obtained lower bound on the run time of this algorithm reproduces $\sqrt N$ scaling ($N$ being the number of database entries) of the explicitly known optimal run time. This observation highlights the merits of the new adiabatic condition and its potential relevance to adiabatic quantum computing.