Deriving generalization bounds for stable algorithms is a classical question in learning theory taking its roots in the early works by Vapnik and Chervonenkis (1974) and Rogers and Wagner (1978). In a series of recent breakthrough papers by Feldman and Vondrak (2018, 2019), it was shown that the best known high probability upper bounds for uniformly stable learning algorithms due to Bousquet and Elisseef (2002) are sub-optimal in some natural regimes. To do so, they proved two generalization bounds that significantly outperform the simple generalization bound of Bousquet and Elisseef (2002). Feldman and Vondrak also asked if it is possible to provide sharper bounds and prove corresponding high probability lower bounds. This paper is devoted to these questions: firstly, inspired by the original arguments of Feldman and Vondrak (2019), we provide a short proof of the moment bound that implies the generalization bound stronger than both recent results (Feldman and Vondrak, 2018, 2019). Secondly, we prove general lower bounds, showing that our moment bound is sharp (up to a logarithmic factor) unless some additional properties of the corresponding random variables are used. Our main probabilistic result is a general concentration inequality for weakly correlated random variables, which may be of independent interest.