A metric space $M$ us said to have the fibered approximation property in
dimension $n$ (br., $M\in \mathrm{FAP}(n)$) if for any $\epsilon>0$, $m\geq 0$
and any map $g: I^m\times I^n\to M$ there exists a map $g':I^m\times I^n\to M$
such that $g'$ is $\epsilon$-homotopic to $g$ and $\dim g'\big(\{z\}\times
I^n\big)\leq n$ for all $z\in I^m$. The class of spaces having the
$\mathrm{FAP}(n)$-property is investigated in this paper. The main theorems are
applied to obtain generalizations of some results due to Uspenskij and
Tuncali-Valov.