  It is known that it is a very restrictive condition for a frame
$\{f_k\}_{k=1}^\infty$ to have a representation $ \{T^n \varphi\}_{n=0}^\infty$
as the orbit of a bounded operator $T$ under a single generator
$\varphi\in\mathcal{H}.$ In this paper we prove that, on the other hand, any
frame can be approximated arbitrarily well by a suborbit $\{T^{\alpha(k)}
\varphi\}_{k=1}^\infty$ of a bounded operator $T$. An important new aspect is
that for certain important classes of frames, e.g., frames consisting of
finitely supported vectors in $\ell^2(\mathbb{N}),$ we can be completely
explicit about possible choices of the operator $T$ and the powers
$\alpha(k),k\in \mathbb{N}.$ A similar approach carried out in
$L^2(\mathbb{R})$ leads to an approximation of a frame using suborbits of two
bounded operators. The results are illustrated with an application to Gabor
frames generated by a compactly supported function. The paper is concluded with
an appendix which collects general results about frame representations using
multiple orbits of bounded operators.
