The answer is yes, the atom does absorb radiation that does not exactly match the transistion frequency. This is due to the Doppler effect that everyone knows from an ambulance with siren driving by. The frequency you hear is higher if the ambulance moves towards you and lower if it drives away from you.
It's the same with the atom. If the atom moves (and it does unless you cool it down to really low temperatures) the observed frequency of the radiation you shine in is shifted depending on the direction of travel of the light and the direction and velocity the atom moves (on the scalar product of both). The phenomen you described is called Doppler broadening of spectral lines and I would say the effect can be described using Bohr's model, since it is a purely classical effect.
The technique to get rid of these broadend lines is called Doppler free spectroscopy. It makes use of some cool techniques you can easily google.
Edit: There are more effects of broadening (like those m0nhawk mentioned in his answer). But under normal conditions the doppler broadening has the biggest effect of all those and overlays the others. The black lines in
Edit 2: Wolfram alpha offers a tool to calculate the above picture are from Dopplerthermal doppler broadening. It says that the line in the picture above ($486\mathrm{nm}$) at $T=300\mathrm{K}$ is broadend $\Delta\lambda\approx 4\cdot10^{-2}Å$