  Let $\bf f$ be a primitive Hilbert cusp form of weight $k$ and level
$\mathfrak{n}$ with Fourier coefficients $c_{\bf f}(\mathfrak{m})$. We prove a
non-trivial upper bound for almost all Fourier coefficients $c_{\bf
f}(\mathfrak{m})$ of $\bf f$. This generalizes the bounds obtained by Luca,
Radziwi\l{}\l{} and Shparlinski. We also prove the existence of infinitely many
integral ideals $\mathfrak{m}$ for which the Fourier coefficients $c_{\bf
f}(\mathfrak{m})$ have the improved upper bound and further we obtain a
refinement of these integral ideals in terms of prime powers. In particular,
this enable us to deduce the bound for Fourier coefficients of elliptic cusp
forms beyond the `typical size'. Moreover, we prove further improvements of the
bound under the assumption of Littlewood's conjecture. Finally, We study a
lower bound for the Fourier coefficients at prime powers provided the
corresponding Hecke eigen angle is badly approximable.
