Frid, Puzynina and Zamboni (2013) defined the palindromic length of a finite word w as the minimal number of palindromes whose concatenation is equal to w. For an infinite word \(\varvec{u}\) we study \(\mathrm {pal}_{\varvec{u}}\), that is, the function that assigns to each positive integer n, the maximal palindromic length of factors of length n in \(\varvec{u}\). Recently, Frid (2018) proved that \(\limsup _{n\rightarrow \infty }\mathrm {pal}_{\varvec{u}}(n)=+\infty \) for any Sturmian word \(\varvec{u}\). We show that there is a constant \(K>0\) such that \(\mathrm {pal}_{\varvec{u}}(n)\le K\ln n\) for every Sturmian word \(\varvec{u}\), and that for each non-decreasing function f with property \(\lim _{n\rightarrow \infty }f(n)=+\infty \) there is a Sturmian word \(\varvec{u}\) such that \(\mathrm {pal}_{\varvec{u}}(n)=\mathcal {O}(f(n))\).