For an odd prime \(p\), let \(I\) by any set of \(m\) consecutive integers contained in \(\{1,\ldots,p-1\}\). The author proves the following result about Bernoulli numbers \(B\): for any rational integer \(\alpha\), the number of \(n\) in \(I\) such that \(\sum^{p-2}_{k=1}B_ kn^{p-1-k}\equiv \alpha \bmod p\), does not exceed \(4.5m^{2/3}\). A related statement about values of Bernoulli polynomials at \(1,\ldots,p-1\) is also proved. An application of the former result was contained in the author's paper [Q. J. Math., Oxf. II. Ser. 37, 257--261 (1986; Zbl 0604.10007)].