Let \(R\) be a left Artinian ring, then the finitistic dimensions of \(R\) are defined as follows: \(\text{lfPD}(R)= \sup\{\text{Pd}(M)\mid M\) a finitely generated left \(R\)-module of \(\text{Pd}(M)< \infty\}\), \(\text{lFPD}(R) =\sup\{\text{Pd}(M)\mid M\) a left \(R\)-module of \(\text{Pd}(M)< \infty\}\), where \(\text{Pd}(M)\) denotes the projective dimension of \(M\). The author determines bounds for the finitistic dimensions of one-sided almost serial rings in terms of the projective dimensions of the simple modules of finite projective dimension and in terms of the number of the simple modules. More precisely, he shows that: 1) For a left almost serial ring \(R\), \(\text{lFPD}(R)\leq\text{Pd}(Rf/Jf)+ 1\), where \(f\) is an idempotent such that the left simple modules appearing in \(Rf/Jf\) are exactly the ones having finite projective dimensions; 2) For a two-sided Artinian left or right almost serial ring, \(\text{lFPD}(R)\leq 2n- 2\), where \(n\) is the number of simple left \(R\)-modules. He also proves that, for an Artinian left uniformly monomial ring \(R\), \(\text{lFPD}(R)= \text{lfPD}(R)\).