The dynamics of polymer translocation through a pore has been the subject of recent theoretical and experimental works. We have considered theoretical estimates and performed computer simulations to understand the mechanism of DNA uptake into the cell nucleus, a phenomenon experimentally investigated by attaching a small bead to the free end of the double helix and pulling this bead with the help of an optical trap. The experiments show that the uptake is monotonic and slows down when the remaining DNA segment becomes very short. Numerical and analytical studies of the entropic repulsion between the DNA filament and the membrane wall suggest a new interpretation of the experimental observations. Our results indicate that the repulsion monotonically decreases as the uptake progresses. Thus, the DNA is pulled in either (i) by a small force of unknown origin, and then the slowing down can be interpreted only statistically, or (ii) by a strong but slow ratchet mechanism, which would naturally explain the observed monotonicity, but then the slowing down requires additional explanations. Only further experiments can unambiguously distinguish between these two mechanisms.