By using an explicit formula for Bernoulli polynomials we obtained in a recent work (in which B_n(x) is written as a linear combination of the polynomials (x - r)ⁿ, r = 1, ... , K + 1, where K ≥ n), it is possible to obtain Bernoulli polynomial identities from polynomial-combinatorial identities. Using this approach, we obtain some generalizations and new demonstrations of the 1971 Carlitz identity involving Bernoulli numbers, and we also obtain some new identities involving Bernoulli polynomials.