There is a nucleus and there is a core--- they mean two different things. You can knock a nucleus out of an atom by smashing it with a heavy particle, or with a fast neutron, or if it undergoes fission, or in many other ways. This is adressed in the other answers.
But the core of an atom is the nucleus plus all the electrons in inner shells. If you were to remove the core of the atom, you would leave behind only the valence electrons, It might seem that it is possible to knock the core out of an atom, because the inner electrons are more tightly bound than the outer electrons by a factor of order unity.
But there is no real mechanism for doing so suddenly without disturbing the atom. The reason is that a fast particle can knock out the nucleus, but it is very unlikely to knock out the remaining core electrons so that they move along with the nucleus--- the phase space is very small.
This has a practical effect--- it means that the energy required to knock a nucleus out of a heavy-atom crystal by collision is much larger than the energy required to completely ionize away the outermost electrons of the atom in vacuum. This means that K-shell vacancies in a heavy metal can only knock out H-isotopes from their lattice positions, where there is no core, not heavy atoms, because to knock out the heavy atom nucleus requires making two K-shell (innermost S-shell) vacancies, and this is at least twice the energy of the K-shell excitation. So although a vacant K-shell has enough energy to knock out the core of the heavy metal atom, the coherent process which does this is unavailable since it requires a conspiracy which simultaneously knocks out the nucleus and the two K-shell electrons at the same time into nearly the same k, and this is a small corner of the outgoing phase space.
This is not really said in the literature, but it is certainly understood in atomic physics. The knock-out interactions are taken particle by particle, and do not involve the atomic cores.
In your question you use "core" to mean "nucleus", so the other answers are more appropriate, but I think it is interesting to adress the question as stated.