Can quantum mechanics model a classical particle in a box?

The classical limit of the particle in a box would indeed be a wave packet with some range $Delta n$ of values. This page discusses this limit in a bit of detail, and provides a handy applet that allows you to play around with various wavepackets and see their evolution in the position and momentum representations.

To summarize in case of link rot: a superposition of states between $n_0-Delta n$ and $n_0+Delta n$, with $Delta n ll n_0$, gives a wavepacket of the type you want. The wavefunction will then be $$ Psi(x,t) = sum_{n=n_0-Delta n}^{n_0+Delta n} C_n sin left( frac{n pi x}{a} right) e^{-i E_n t/hbar} $$ The coefficients $C_n$ for the $n$th eigenstate are somewhat arbitrary; we choose them to be $$ C_n propto cos^2 left[frac{(n- n_0) pi}{2 n_0 + 1} right] e^{-i (n-n_0)pi/2} $$ (Simply replacing the cosine above with a constant will yield similar results but the initial wavepacket won't be as smooth.) This leads to a "smooth" wavepacket that "bounces around" inside the box for some time. However, it also disperses with time (i.e., $Delta x$ increases), as we would expect from a free particle. The wave packet also interferes with itself substantially when it bounces off of the "walls" of the box.