Summary: We study certain classes of exceptional times of a linear Brownian motion \((B_t\), \(t \geq 0)\). In particular, we consider the set \(K^-\) of all instants \(t \in [0,1]\) such that the value \(B_t\) of the Brownian motion at time \(t\) is greater than its mean value over all intervals \([s,t]\), \(s < t\). We also study the subset \(K\) of \(K^-\) of all instants \(t\) such that in addition \(B_t\) is greater than the mean value of \(B\) over the intervals \([t,s]\), \(t < s \leq 1\). We compute the Hausdorff dimension of \(K^-\), \(K\) and some other related sets of exceptional times. These results are closely related to a recent work of Y. G. Sinaj [Commun. Math. Phys. 148, No. 3, 601-621 (1992; Zbl 0755.60105)] motivated by the analysis of solutions to the Burgers equation with random initial data. The proofs involve studying suitable approximations of the sets \(K^-\) and \(K\), and deriving precise estimates for the probability that a given time \(t\) belongs to these approximations. A delicate zero-one law argument is also needed to prove that the lower bound on the dimension of \(K\) holds with probability 1.